advances in numerical modelling of adhesive jointsadvances in numerical modelling of adhesive joints...

Advances in Numerical Modellingof Adhesive Joints

Abstract The analysis of adhesively bonded joints started in 1938 with theclosed-form model of Volkersen. The equilibrium equation of a single lap joint ledto a simple governing differential equation with a simple algebraic equation.However, if there is yielding of the adhesive and/or the adherends and substantialpeeling is present, a more complex model is necessary. The more complete is ananalysis, the more complicated it becomes and the more difficult it is to obtain asimple and effective solution. The finite element (FE) method, the boundaryelement (BE) method and the finite difference (FD) method are the three majornumerical methods for solving differential equations in science and engineering.These methods have also been applied to adhesive joints, especially the FEmethod. This book deals with the most recent numerical modelling of adhesivejoints. Advances in damage mechanics and extended finite element method aredescribed in the context of the FE method with examples of application. Theclassical continuum mechanics and fracture mechanics approach are also intro-duced. The BE method and the FD method are also discussed with indication ofthe cases they are most adapted to. There is not at the moment a numericaltechnique that can solve any problem and the analyst needs to be aware of thelimitations involved in each case.

Keywords Adhesive joints � Finite element method � Continuum mechanics �Fracture mechanics � Damage mechanics � Extended finite element method �Boundary element method � Finite difference method

1 Introduction

Adhesive bonding is a material joining process in which an adhesive, placedbetween the adherend surfaces, solidifies to produce an adhesive bond. Adhesivelybonded joints are an increasing alternative to mechanical joints in engineering

L. F. M. da Silva and R. D. S. G. Campilho, Advances in Numerical Modellingof Adhesive Joints, SpringerBriefs in Computational Mechanics,DOI: 10.1007/978-3-642-23608-2_1, � Lucas F. M. da Silva 2012

1

applications and provide many advantages over conventional mechanical fasten-ers. Among these advantages are lower structural weight, lower fabrication cost,and improved damage tolerance. The application of adhesively bonded joints instructural components made of fibre reinforced composites has increased signifi-cantly in recent years. The traditional fasteners usually result in the cutting offibres and hence the introduction of stress concentrations, both of which reducestructural integrity. By contrast, bonded joints are more continuous and havepotential advantages of strength-to-weight ratio, design flexibility and ease offabrication. In fact, adhesive bonding has found applications in various areas fromhigh technology industries such as aeronautics, aerospace, electronics, and auto-motive to traditional industries such as construction, sports and packaging. Thereare several reference books dealing with adhesive joints such as those of Adamset al. (1997), Kinloch (1987) and more recently that of da Silva et al. (2011).

Bonded joints are frequently expected to sustain static or cyclic loads forconsiderable periods of time without any adverse effect on the load-bearingcapacity of the structure. A lack of suitable material models and failure criteria hasresulted in a tendency to ‘overdesign’ adhesive joints. Safety considerations oftenrequire that adhesively bonded structures, particularly those employed in primaryload-bearing applications, include mechanical fasteners (e.g. bolts) as an addi-tional safety precaution. These practices result in heavier and more costly com-ponents. The development of reliable design and predictive methodologies can beexpected to result in more efficient use of adhesives. In order to design structuraljoints in engineering structures, it is necessary to be able to analyse them.This means to determine stresses and strains under a given loading, and to predictthe probable points of failure. There are two basic mathematical approaches for theanalyses of adhesively bonded joints: closed-form analyses (analytical methods)and numerical methods (i.e. finite element or FE analyses). Differential equationsare derived by applying a physical principle such as conservation of mass,momentum or energy. These equations govern the kinematic and mechanicalbehaviour of general bodies. The analysis of adhesively bonded joints started70 years ago with the simple closed-from model of Volkersen (1938) that con-siders the adhesive and adherends as elastic, and that the adhesive deforms only inshear. The equilibrium equation of a single lap joint led to a simple governingdifferential equation with a simple algebraic equation. However, the analysis ofadhesive joints can be highly complex if composite adherends are used, theadhesive deforms plastically or if there is an adhesive fillet. In those cases, severaldifferential equations of high complexity might be obtained (non-linear and non-homogeneous). For those cases, numerical methods are more adequate. The FEmethod, the boundary element (BE) method and the finite difference (FD) methodare the three major numerical methods for solving partial differential equations inscience and engineering. The FE is by far the most common technique used in thecontext of adhesively bonded joints. Adams et al. are among the first to have usedthe FE method for analyzing adhesive joint stresses (Adams and Peppiatt 1974;Crocombe and Adams 1981; Adams and Harris 1984; Adams et al. 1986; Adamsand Davies 2002). One of the first reasons for the use of the FE method was to

2 Advances in Numerical Modelling of Adhesive Joints

assess the influence of the spew fillet. The joint rotation and the adherends andadhesive plasticity are other aspects that are easier to treat with a FE analysis.The study of Adams and Harris (1984) is one of the first FE analyses taking intoaccount these three aspects. The use of the BE method is still very limited in theanalysis of adhesive joints. The FD method is especially used for solving complexgoverning differential equations in closed-form models. The three numericaltechniques are treated in three separate sections. The book was prepared forengineers and scientists that have already some background in adhesive joints andnumerical modelling. A brief description of each numerical method is given andthe most recent advances made concerning the analysis of adhesive joints arediscussed. The FE is obviously the one treated in more detail due to its importance,and several approaches to failure analysis are accounted for: continuummechanics, fracture mechanics and the more recent damage mechanics andextended finite element method (XFEM). Examples of application of damagemechanics and XFEM are given to illustrate the most recent advances in numericalmodelling of adhesively bonded joints.

2 Finite Element Method

The FE method is a numerical analysis procedure that provides an approximatesolution to problems in various fields of engineering. Ashcroft (2011) gives adescription of the method applied to adhesive joints. The FE method is based onthe idea of building a complicated object with simple blocks or dividing a com-plicated object into small and manageable pieces. The first efforts to use piecewisecontinuous functions defined over triangular domains appeared in applied math-ematics literature with the work of Courant (1943). Advances in the aerospaceindustry and the development of computers in the 1950s and 1960s saw furtherdevelopment and computerization of these methods (Turner et al. 1956). This wasthe direct pre-cursor of nearly all current commercial FE analysis methods. Theterm FE was introduced by Clough (1960). Zienkiewicz and Cheung (1967)describe the applicability of the method to general field applications. In the 1980s,generalised software packages designed to run on powerful computers weredeveloped and improved techniques for the analysis of non-linear problems wereestablished. The FE method is now used in practically all fields of engineeringanalysis such as structural, heat transfer, material transport (such as diffusion),fluid mechanics and electro-magnetics. Recent FE programs offer the possibility tomulti-physics problems (coupling of different analysis classes, such as thermo-structural or hygro-thermo-structural problems).

To predict the joint strength, one must have the stress distribution and a suitablefailure criterion. The stress distribution can be obtained by a FE analysis or aclosed-form model. For complex geometries and elaborate material models, the FEmethod is preferable. One of the simplest failure models is that based on a stress orstrain limit state, i.e. based on a continuum mechanics approach. Fracture

1 Introduction 3

mechanics principles can also be used within a FE analysis. This can be based oneither the stress intensity factor or energy approaches. An extension to thisapproach is damage modelling with cohesive zone elements or continuumelements, which also allows to account for damage of a material ahead of a crack.This technique is a combination of a continuum mechanics and fracture mechanicsapproach. Another method for modelling cracks in materials is the XFEM, whichuses enriched shape functions to represent a discontinuous displacement field. Themain advantage of XFEM is that the crack may initiate at any point in the materialand propagate based on loading conditions. No remeshing is required as the crackcan grow within an element, and it does not need to follow element boundaries. Allthese approaches (continuum mechanics, fracture mechanics, damage modellingand XFEM) are described in this section with examples of application, especiallyfor the most advanced methods of damage modelling and XFEM. However, a briefdescription of the FE method is first given with simple guidelines for modellingadhesive joints.

2.1 Simple Description

Basically, this method involves the discretization of a structure in various subdomains, known as elements, joined at their nodes. Each node has a limitednumber of degrees of freedom (dof). Hence, the continuum is now represented by afinite number of dof, determined by the number of elements, the number of nodesper element and the number of dof per node. In an element, the field quantity orbehaviour of the model (for example displacements, d, in stress analyses) isinterpolated from the values of the field quantity at the nodes. Connecting theelements by the nodes, the field quantity can be interpolated over the wholestructure. The most common formulation method in FE analysis is the variationalmethod. This method involves the solution of a governing partial differentialequation (for example the equilibrium equations in elasticity problems), bydetermining the conditions that make a functional stationary, i.e., maximum orminimum. In elasticity problems, the functional used is the total potential energyof the structure. The optimal values of the field quantity are those that minimisethe total energy of the system, satisfying internal compatibility and essentialboundary conditions. The process of minimisation creates a system of algebraicequations for the field quantity at the nodes. The matrix symbolism used for thatsystem of equations is

Kd ¼ F; ð1Þ

where d is a vector with the values of the field quantity at the nodes (values of d forexample), F is a vector of known loads (for example forces in elasticity problems)and K is a matrix of known constants that represents the property of the elements.In elasticity problems, K is the stiffness matrix. In stress analyses, to simulate a


given loading on the model, boundary conditions and loads or values of d areapplied and the output values of d are obtained from Eq. 1. However, the stiffnessmatrix K in Eq. 1 contains integrals that generally cannot be solved algebraically.Instead, a numerical integration scheme (such as Gauss quadrature) has to be usedto determine the stiffness matrix. In general, the integrals can be computed bysampling from a number of points (Gauss points) and multiplying by an appro-priate value or weighting factor. As the number of Gauss points increases, moreaccurate integration results are typically obtained. However, using too many Gausspoints would require more computational resources and the results may notimprove. The numerical solution of Eq. 1 gives the d values, which in turn allowthe calculation of strains and stresses along the whole structure. For more infor-mation on the FE method, the reference books of Cook (1995) for a first intro-duction and that of Zienkiewics and Taylor (2001) for a more detailed descriptionare recommended.

The FE method allows studying any type of geometry, i.e., it can take intoaccount variations in the adherends shape and the adhesive fillet, which is verydifficult or impossible with the closed-form analyses. Another important advantageof the method is that it enables to calculate all the stress and strain components of astructure for more realistic strength predictions. However, in general parametricFE studies (e.g. study of a geometrical parameter such as the adhesive thickness,tA) are more difficult than with closed-form models because they generally involvecreating a new model for each new configuration.

The commercial FE programs permit to easily include geometric non-linearitiessuch as those occurring in the single lap joint. There are also a variety of materialmodels from linear elastic elastic to visco-plastic. To determine the initial yieldingand subsequent plastic deformation in a bonded joint, a yielding model needs to beincluded for the adhesive and possibly for the adherend. For metallic substrates,the von Mises yield criterion may be applied. For composite adherends, yieldingseldom occurs and failure criteria are used instead. In the case of adhesives andpolymeric adherends, a yielding model that takes into account the hydrostaticpressure is generally required such as that proposed by Raghava et al. (1973).Failure is discussed in more detail in the following sections.

As seen above, the domain or structure is divided in elements leading to a mesh.Stress concentrations need a fine mesh to capture the stress gradients. In theparticular case of stress singularities, mesh refinements may lead to convergenceproblems since the stress tends to infinity as finer elements are used. Elasto-plasticanalyses or fracture mechanics concepts are advised in those cases, as discussed inthe coming sections. A pre-processor is used to generate the geometry and themesh. It is advised to apply the boundary conditions and the loads so that the meshcan be easily modified without altering the boundary conditions, to reduce themeshing effort. The geometry should be modelled with precision using at the sametime any symmetry that reduces the size of the model and the number of calcu-lations. The boundary conditions should simulate the reality as closely as possible.The most common boundary conditions for single lap joints are those representedin Fig. 1, which simulate gripping and loading in a testing machine. When double

2 Finite Element Method 5

lap joints are analysed, half of the joint is sufficient for analysis due to symmetry,as shown in Fig. 2.

Various factors should be taken into account when preparing a mesh for abonded joint, such as the mesh density and the type of element. The choice ofelements (for example simple beams and continuum solid elements) has animportant effect on the final results. Each type of element has advantages anddisadvantages depending on the application. Continuum solid elements are mostsuited for linear analyses and also for complex non-linear problems involvingplasticity and large deformations. An adhesively bonded joint can be modelled intwo dimensions (2D) (Figs. 1 and 2) or three dimensions (3D). The 2D con-tinuous elements include plane stress, plane strain and generalised plane strain.The plane stress elements are used when one of the dimensions of the body isvery small in relation to the others. This situation occurs in plates where thestresses in the thickness direction are considered nil. The plane strain elementsare used when the width is much larger than the thickness. This is the case of asingle lap joint where the strains in the width direction are considered nil.In the case of the generalised plane strain elements, two parallel rigid planesexist that can only move away or closer to each other, which permits to accountfor the transversal strains. Continuous 3D elements suppress the approximationintroduced by the plane stress or plane strain conditions. Despite a 3D analysisgiving more accurate results than 2D analyses, the time and effort of a 3Danalysis are often not justified.

Fig. 1 Finite element mesh and boundary conditions for a single lap joint

Fig. 2 Boundary conditions for a double lap joint


In adhesively bonded joints, a 2D analysis in plane strain conditions is oftenpreferred. Despite 3D effects such as the lateral deformation (Adams andPeppiatt 1973) and the anticlastic bending (Adams and Davies 1996; Gonçalveset al. 2002), various studies have shown that 2D analyses give accurate enoughresults (Adams et al. 1997; Adams and Davies 2002). The elements are oftenisoparametric quadratic elements of eight nodes. Isoparametric triangular ele-ments of six nodes are also used, especially in areas that need mesh refinement.Generally, a first coarse mesh is modelled and the mesh is refined progressively(increasing the number of elements by decreasing their size). A reduction of theelement size increases the stress and strain levels until a point where there ispractically no change, in which a converged mesh is attained. In the case themodel contains a singularity, which is common in adhesive joints, there is noconvergence and the values tend to infinity as the mesh is refined. Afterthe process step where the calculations are made by the computer, the resultsare analysed in the postprocess step. The first thing to do is to check if theimposed loading and boundary conditions create a logical deformed shape.The continuity of the stress contours must also be checked to minimise anydiscontinuity.

Figure 3 illustrates the 2D deformed shape of a single lap joint (a) and thepeel and shear stress distributions in the adhesive bond along the overlap, nor-malized by the average shear stress in the bond (b) for a joint with carbon-epoxy

-2

0

2

4

6

y/

avg

, xy

/av

g

Normalized Overlap

peel ( y) shear ( xy)

0 0.2 0.4 0.6 0.8 1

(a)

(b)

Fig. 3 Deformed shape of a single lap joint at the overlap (a) and normalized peel and shearstress distributions in the adhesive bond along the overlap (b) (Campilho 2009)


adherends and a ductile adhesive bond (Campilho 2009). In the areas where thestress is relatively uniform, a coarser mesh can be used to reduce the compu-tation time. An example of a mesh is given in Fig. 1. The mesh is usually refinedat the ends of the overlap where stress concentrations exist due to the sharpgeometry change. The adherends mesh is also refined along the overlap. Thecommercial programs have pre-processors which facilitate the mesh generationof complicated shapes.

The modelling of laminated composite materials requires the definition of localcoordinates to guarantee that the orientation of the layers is coincident with themodel orientation (Fig. 4). For curved sections, the orientation of the system mustbe coincident with the laminate orientation.

The numerical problem can be solved using implicit or explicit mathematicalcodes. The implicit code is used for solving a great variety of linear and non-linearproblems and it is based on the resolution of Eq. 1. The solution is consideredacceptable when the difference between the two members of the equation is lowerthan a value (residue) that is determined by the user. Therefore, the processrequires an iterative procedure for a given solution in each load increment. Theexplicit code is adequate for short transient dynamic events such as impactloadings, and is also very efficient for highly non-linear problems. The explicitmethod is particularly useful for modelling ductile adhesives that exhibit highfailure deformations. It needs very small increments in static analyses so that thekinetic energy is always negligible. The term convergence is used to indicate thatthe process used to solve the equations system converges, and the solution at theend of an increment is a solution that by definition converged. If a solution has notbeen found for a given increment, the method reduces the size of the incrementand solves again the equation. If a too small increment is reached withoutconvergence, it is considered that the analysis method and the model have notconverged. Convergence problems are generally associated to materials thathave an unstable behaviour and occur frequently in non-linear problems. Thenon-convergence is also less common in loadings applied by a specified value of dthat by a force.

Fig. 4 Local coordinates in the case of laminated composites


2.2 Continuum Mechanics Approach

In the continuum mechanics approach, the maximum values of stress, strain orstrain energy, predicted by the FE analyses, are usually used in the failure criterionand are compared with the corresponding material allowable values. Initially, themaximum principal stresses were proposed for very brittle materials whose failuremode is at right angles to the direction of maximum principal stress. This criterionignores all the other principal stresses, even though they are not nil. Establishingthe failure modes in lap joints bonded with brittle adhesives, Adams et al. (1997)have extensively used this criterion to predict joint strength with success.However, because of the singularity of stresses at re-entrant corners of joints, thestresses depend on the mesh size used and how close to the singular points thestresses are taken. Values of stresses calculated at Gauss points near the singularityor extrapolation of Gauss point values to the singularity were, in fact, used.Therefore, care must be taken when using this criterion. Although the criterion issensitive to the mesh size used, the physical insight into the failure process is veryclear, as the maximum principal stress is the most responsible for the failure ofjoints bonded with brittle adhesives. However, it should be noted that the adherendcorners are usually not sharp in practice. There is, in general, a small amount ofrounding at the adherend corner due to the production process. This may affect thestress distributions in the region of the adherend corner and, therefore, the jointstrength, because stresses in this area are very sensitive to the change in thegeometry. One consequence of the adherend rounding is the non-existence of thesingularity, which facilitates the application of a stress or strain limit criterion.Adams and Harris (1987) theoretically and experimentally demonstrated that thestrength of single lap joints with rounded adherends with a toughened adhesiveincreased substantially compared with joints with sharp adherend corners. Morerecently, Zhao et al. (2011a, b) have also studied the effect of adherend rounding(Fig. 5). An example of stress distribution is given in Fig. 6 as a function of thedegree of rounding, showing that the stress singularity vanishes with a smalldegree of rounding.

Von Mises proposed a yield criterion, which states that a material yields undermulti-axial stresses when its distortion energy reaches a critical value, that is

r2VM ¼ r1 � r2ð Þ2þ r2 � r3ð Þ2þ r3 � r1ð Þ2¼ constant, ð2Þ

where ri (i = 1, 2, 3) are the principal stresses. Such a criterion has been used byIkegami et al. (1990) to study the strength of scarf joints between glass fibrecomposites and metals. It should be noted that this criterion is more applicable tomaterial yielding than strength.

Shear stresses have been extensively used to predict lap joint strength, espe-cially in closed-form analyses, considering a limiting maximum shear stress equalto the bulk adhesive shear strength. These are also described here for a completedescription of the continuum mechanics approach. Greenwood (1969) used the


maximum shear stress calculated by Goland and Reissner’s analysis (Goland andReissner 1944) to predict joint strength. The Engineering Sciences Data Unit(ESDU 1979) implemented this criterion into a commercial package. Morerecently, John et al. (1991) used shear stresses together with a critical distance to

Fig. 5 Aluminium/epoxy single lap joints with different degrees of rounding (Zhao et al. 2011a)

Fig. 6 Maximum principal stresses in the adhesive with a 20 kN applied load, close to theunloaded adherend (Zhao et al. 2011a)


predict the strength of double lap joints. Lee and Lee (1992) also used the max-imum shear stress in tubular joints. da Silva et al. (2009a, b) showed for single lapjoints that this criterion is only valid for brittle adhesives and short overlaps. Thisapproach ignores the normal stresses existing in lap joints and therefore it over-estimates the joint strength.

When ductile adhesives are used, criteria based on stresses are not appropriatebecause joints can still endure large loads after adhesive yielding. For ductileadhesives, Adams and Harris (1984) used the maximum principal strain as failurecriterion for predicting the joint strength. This criterion can also predict the failuremode. However, it is equally sensitive to the mesh size, as previously discussed forthe maximum principal stress approach.

Hart-Smith (1973) proposed that the maximum shear strain might be used as afailure criterion when plastic deformation was apparent. da Silva et al. (2009c)implemented this criterion and others into a commercial package. Other analysesgo beyond that of Hart-Smith, which allows both shear and peel contributions toplasticity such as that by Adams and Mallick (1992). ESDU (1979) also imple-mented the maximum shear strain criterion in their commercial program. da Silvaet al. (2009a, b) have shown, for single lap joints, that the maximum shear straincriterion is very accurate for ductile adhesives.

Clarke and McGregor (1993) predicted failure when the maximum principalstress exceeded the maximum uniaxial stress for a bulk adhesive over a certainlength normal to the direction of the maximum principal stresses. No justificationwas given for the choice of zone size. It was noted that the sensitivity to changes inlocal joint geometry, such as radius of the adherend corner, was reasonably low forthis criterion. Crocombe (1989) studied the failure of cracked and uncrackedspecimens under various modes of loading and used a critical peel stress at adistance from the singularity with some success. An alternative method was alsoproposed to use an effective stress, matched to the uniaxial bulk strength, at adistance. However, it was found for the latter criterion that the critical distance atwhich it should be applied varied with different modes of loading because of thechange in the plastic zone size. No general criterion for a given adhesive waspresented. Kinloch and Williams (1980) and Kinloch and Young (1983) alsoconsidered some cracked specimens, and applied failure criteria at criticaldistances with some success, but the work was not extended to considerun-cracked continua. There is no real physical justification for these criteriaapplied at a distance, and many of them are dependent on parameters such as tA,which means that no general criterion of failure is available within these methods.

The strain energy is the area under the stress–strain (r–e) curve. Therefore, bothstress and strain criteria can be related to strain energy. However, it should benoted that criteria based on strain energy take account of all the stress and straincomponents. As a result, they are more suitable as a failure criterion than eitherstresses or strains alone. Plastic energy density has also been used as a failurecriterion (Adams and Harris 1987), being similar to the total strain energy criterionbut it only takes the plastic part of the deformation into account. Zhao et al.(2011b) applied a criterion whereby if the average plastic energy density over a


certain distance within the single lap joint reached a critical value, then the jointwas deemed to have failed. The specific energy is not so sensitive to the size of theintegration zone, as it is ‘averaged’ over an area (2D analyses) or volume (3Danalyses). It is common knowledge that the accuracy of the FE approach is morereliable when it is interpreted as an average, rather than in a pointwise sense. Theintegration region is usually chosen as a whole element for numerical convenience.In Zhao et al. (2011b), the value of tA was used for integration and the predictionscompared very well with the experimental results for a ductile adhesive.

In the analysis of lap joints, Crocombe (1989) found that, for largely ductileadhesives, the whole overlap yielded before failure. A new failure criterion wasthen proposed based on the yielding of adhesive in the whole overlap. Once a pathof yielding was found in the overlap with a given load, the joint was thought to befailed. Such a criterion is useful for very ductile adhesives in which the adhesivebond cannot support any larger load once it yields globally. However, it should benoted that the adhesives need to be extremely ductile (more than 20% of failurestrain in shear) for the whole adhesive bond to yield before final failure (Adamset al. 1997; da Silva et al. 2008; da Silva et al. 2009d). Also, this criterion is onlyapplicable to lap joints. Unfortunately, for certain geometric conditions joints tendto fail before the whole adhesive bond yields.

It should be realized that all the above criteria are applicable to continuousstructures only. They run into difficulty when defects occur or more than onematerial is present, since stresses or strains are not well defined at the singularpoints. As a result, new criteria or modified versions of the above criteria need tobe developed.

2.3 Fracture Mechanics Approach

Continuum mechanics assumes that the structure and its material are continuous.Defects or two materials with re-entrant corners obviously are not consistent withsuch an assumption. Consequently, continuum mechanics gives no solution atthese singular points resulting in stress or strain singularities. Cracks are the mostcommon defects in structures, for which fracture mechanics has been developed.In fracture mechanics, it is well accepted that stresses calculated by using con-tinuum mechanics are singular (infinite) at the crack tip. The reason why singu-larities exist is explained as follows. Figure 7 shows stresses around the tip of asharp crack in an infinitely large plate given by continuum mechanics. Physically,the y-stresses, ry, at the crack tip A must be finite (instead of infinite as theorypredicts). However, ry stresses into the crack and away from the tip of the crack(shown as B in Fig. 7) are nil because of the free surfaces. Consequently, adiscontinuity of ry is apparent at point A unless ry is zero there. Such a stressdistribution cannot be accommodated in continuum mechanics, which requires allthe stresses to be continuous. As a result, stresses at the crack tip are not defined(being infinite).


With current theories on mechanics, such a singularity always exists when thecrack angle is \1808. This result was found by Williams (1959) for stresssingularities in a wedged notch. This argument is also applicable to the stresssingularity in two materials bonded together with a re-entrant corner. Actually, thestress discontinuity still exists, although the free surfaces do not exist.

For ductile materials, a large amount of material yielding occurs and the crackmay propagate stably before final failure. Thus, linear elastic fracture mechanics(LEFM) does not work anymore for such materials. The HRR (Hutchinson RiceRosengreen) solution developed by Hutchinson (1968) and by Rice and Rosengren(1968) has, however, been extensively used in ductile fracture. Another importantparameter governing failure is the so-called crack tip opening displacement(Dugdale 1960). However, a strain singularity still exists for ductile materials,even though the stress singularity has disappeared.

Fracture mechanics has been successfully applied to many engineering prob-lems in recent years. The damage tolerance design concept, originally adopted inthe aircraft industry, was based mainly on the well-established concept of LEFM,and it has gradually gained ground in other engineering fields. Many studiesdealing with adhesive joints use the strain energy release rate, G, and respectivecritical value or fracture toughness, Gc (Fernlund and Spelt 1991; O’Brien et al.2003; Cheuk et al. 2005; Shahin and Taheri 2008) instead of stress intensity factorsbecause these are not easily determinable when the crack grows at or near to aninterface. However, the fracture of adhesive joints inherently takes place undermixed mode because of the varying properties between different materials and thecomplex stress system. Failure criteria for mixed mode fracture can be developedin a way analogous to the classical failure criteria, although the fracture surface(or envelope) concept must be introduced. Various mathematical surface functions

Fig. 7 Stress discontinuityaround (a) a crack tip and(b) a re-entrant corner


have been proposed to fit the experimental results, such as the 3D criterion (Dillardet al. 2009):

Gn

Gcn

� �a

þ Gs

Gcs

� �b

þ Gt

Gct

� �c

¼ 1; ð3Þ

where Gn, Gs and Gt are the values of G under pure tension, shear and tearingmodes, respectively, and Gc

n, Gcs and Gc

t the respective values of Gc. The linearenergetic criterion (a = b = c = 1) and the quadratic one (a = b = c = 2) arethe most used. The law parameters may be chosen to best fit the experimental data,or they may be prescribed based on some assumed relationship. Alternate formsfor fracture envelope criteria have also been proposed (Kinloch 1987; Hashemiet al. 1989; Charalambides et al. 1992). In all cases, the failure surface can bemade to match experimental results very closely, by including additional con-stants. Constructing such a surface will require an increasing number of inde-pendent material parameters, which must all be best fitted by experiments. Apartfrom the amount of experimental work involved, this purely mathematical surfacefitting method does not help much to understand the physical failure mechanism ofmixed mode fractures. Although fracture mechanics is mainly used for dealingwith sharp cracks, angular wedged notches are also of practical importance,as shown in Fig. 8 for the case of adhesive joints.

The use of a generalized stress intensity factor, analogous to the stress intensityfactor in classical fracture mechanics, to predict fracture initiation for bondedjoints at the interface corners has also been investigated (Xu et al. 1999). Groth(1988) assumed that initiation of fracture occurs when the generalized stressintensity factor reaches its critical value, initially tuned experimentally. Gleichet al. (2001) carried out a similar study by calculating the singularity strength andintensity for a range of tA values. These approaches work well for the joints that

Fig. 8 Singularities in adhesive lap joints with different spew fillet geometry


were used to determine the critical stress intensity factor but their application isquestionable for extrapolation to other types of geometries. Fracture mechanicscan thus be used to predict joint strength or residual strength if there is a crack tipor a known and calibrated singularity (Clarke and McGregor 1993).

When materials deform plastically, the LEFM concepts have to be extendedinto elasto-plastic fracture mechanics. The J-integral proposed by Rice andRosengren (1968) is suited to deal with these problems. The analytical definitionof the J-integral is given by Rice as an energy line integral

J ¼ZC

w dx2 � Tjouj

ox1dS; ð4Þ

where C denotes a curve surrounding the crack tip, the variable S indicates thearclength, w is the energy density, Tj is the traction vector, uj is the displacementvector and x1 – x2 is the coordinate system. The integral path can be arbitraryprovided that it travels from one of the crack surfaces to another counter-clockwiseand it encloses the crack tip. The J-integral approach has been used by a variety ofresearchers to predict the joint strength of cracked adhesive joints with goodresults (Fernlund et al. 1994; Choupani 2008; Sørensen and Jacobsen 2003; Baneaet al. 2010). Nonetheless, for ductile adhesives the value of Gc is not independentof the joint geometry (Kinloch and Shaw 1981; Hunston et al. 1984). This ismainly because the adherends restrict the development of the yield zone in theadhesive bond, making Gc a function of joint geometry. Chen et al. (2011a) haveshown that the vectorial J-integral is interface length dependent for a bondedbi-material system, as shown in Fig. 9.

Paths 8, 9 and 10 correspond to the same interface length (10.8 mm). The twocomponents of the vector J were considered in the analysis. The classical intro-duction of the J-integral by Rice focused the attention on the J1 component whichis responsible for controlling the forward propagation of the crack. However, in thecontext of multi axial loading and crack turning it is more appropriate to considerboth components of the vector J-integral, since more exhaustive information isprovided for the application of various crack path determination criteria.

The major advantage of the path-independence of the J-integral in homoge-neous materials disappears due to the interface length dependence and practice ofusing far field stresses and strains to calculate J is no longer possible. Under theseconditions, the J-integral must be extrapolated against interface length to a point sothat a great deal of mesh refinement is required. In real joints, the bondline isusually very thin and the two interfaces with the adherends will interact with eachother. The interference of different singular sources so closely sited (thin bond-lines) makes numerical extrapolation very difficult. If an integral path includesboth interfaces, some singular effects will offset each other and the accuracy isthen doubtful. Therefore, the parameter J may not be used as a strength criterionfor joints without a pre-crack. Moreover, failure modes are extremely complicatedin the area of stress or strain concentrations. Therefore, if fracture mechanics is tobe used for ductile adhesives, the process will be prohibitive: first it needs to be


Fig. 9 Elastic J-integral 1and 2 against path length for atwo-material bond undertension. a Bi-material model;b integral J2 as a function ofthe path length; c Integral J1

as a function of the pathlength (Chen et al. 2011a)


determined when and where a crack with a reasonable size initiates. Then, thiscrack has to propagate stably and be arrested. Next, a new crack at another placemay initiate and propagate into an unstable one, causing catastrophic failure. Thisprocess is very difficult to simulate because the crack sizes and positions are verydifficult to determine. Furthermore, such a process will involve a large amount ofcomputing time with very fine mesh, and the mesh shifting and releasing involvedin crack propagation will be very difficult to perform.

2.4 Damage Mechanics

Advanced modelling techniques are required that comprise accurate failure pre-dictions, surpassing the aforementioned limitations associated to the continuumand fracture mechanics approaches, to effectively model damage evolution withina material or structure with bonded components (Liljedahl et al. 2006). Structuraldamage during loading can be found in the form of micro-cracks over a finitevolume or interfacial region between bonded components, such that load transferis locally reduced, globally resulting on a drop of applied load for a given value ofd applied to the structure. Figure 10 reports on a typical uni-axial r–e diagram upto failure for a ductile material or structure. A FE model built solely with solidcontinuum elements comprising the elastic and plastic constitutive behaviours ofeach one of the materials wrongly gives as modelling output the abcd0 curvebecause of generalised plasticization without damage evolution, while a damageand failure based model can actually provide the real abcd curve, by allowingdamage to grow through the simulation of material stiffness degradation betweenpoint c (damage onset) and point d (complete failure).

Damage mechanics permits the simulation of step-by-step damage and fractureat a pre-defined crack path or arbitrarily within a finite region up to completestructural failure (Duan et al. 2004). However, this is still an innovative field underintense development, regarding more accurate modelling techniques, reliable andsimple parameter determination methods, increase of robustness and eliminationof convergence issues (Liljedahl et al. 2006), and it is also under heavyimplementation in commercial FE software packages such as Abaqus� (Campilhoet al. 2011a). The available techniques for damage modelling can be separated intolocal or continuum approaches. In the local approach, damage is confined to a zerovolume line or a surface, allowing the simulation of an interfacial failure betweenmaterials, e.g. between the adhesive bond and the adherend (Yang et al. 2001), theinterlaminar failure of stacked composites (Turon et al. 2007a) or the interfacebetween solid phases of materials (Chandra et al., 2002). By the continuumapproach, the damage is modelled over a finite region, within solid finite elementsof structures to simulate a bulk failure (Song et al. 2006) or along an adhesive bondto model a cohesive fracture of the adhesive bond (Kafkalidis and Thouless 2002).The use of cohesive zone models (CZM’s) coupled to conventional FE analyses isthe most widespread method of predicting static or fatigue damage uptake in


structures (Tvergaard and Hutchinson 1992, Tvergaard and Hutchinson 1993, Weiand Hutchinson 1997, Wei and Hutchinson 1999, Yang et al. 1999, Yang et al.2000, Cavalli and Thouless 2001, Yang and Thouless 2001, Campilho et al. 2005,Campilho et al. 2007). CZM’s are based on the concepts of stress and fracturemechanics, and can be fitted into the local or continuum approach, since they caneither be considered to model the interfacial fracture behaviour of equally ordifferently oriented plies in stacked composites or the adhesive/adherend interfaceto simulate adhesive failures (local approach), or on the other hand to simulate athin bulk layer of a constant thickness material (continuum approach). Under thecontinuum assumption, thin adhesive bonds to join structural members are a largefield of application of CZM’s (Campilho et al. 2008a), but the single row ofcohesive elements used to model the bulk strip of adhesive makes it impossible todifferentiate thickness-wise effects or concentrations of stresses towards theinterface (Magalhães et al. 2005), providing an equivalent behaviour of the bond.Differences exist regarding the definition of the CZM laws, namely in whichconcerns their initial stiffness, between the local approach modelling (the initialstiffness can use the penalty function method) and continuum CZM modelling (theinitial stiffness must accurately reproduce the deformation behaviour of the thinmaterial strip). These issues will be addressed in detail in Sect. 2.4.1 and‘‘Triangular Cohesive Zone Model’’. As CZM modelling is nowadays largelypreponderant for researchers and designers, both in terms of modelling develop-ments and validation/application studies of these methodologies and, additionally,it is very much alike between the local or continuum formulations, the damagemechanics section (Sect. 2.4) is separated into CZM modelling and alternativeapproaches to CZM modelling, the former with much more detail due to theaforementioned reasons, and the latter comprising alternative continuum damagemodelling (ACDM) techniques to CZM formulations, equally tested in recentyears under the scope of bonded structures.

2.4.1 Cohesive Zone Modelling

Although the computer implementation of LEFM techniques endured greatsuccess some decades ago, they are restricted to small-scale yielding beyond thecrack tip. Since for modern toughened adhesives the plastic zones developing alongthe adhesive bond can be larger than the adherends thickness, tP (Ji et al. 2010),

Fig. 10 Typical uni-axialr–e response of a ductilematerial or structure


a large effort was undertaken beginning in the late 1950s/early 1960s by the inde-pendent studies of Barenblatt (1959, 1962) and Dugdale (1960). The concept ofcohesive zone was proposed by these authors to describe damage under static loadsat the cohesive process zone ahead of the apparent crack tip. CZM’s were largelyrefined and tested since then to simulate crack initiation and propagation in cohesiveand interfacial failure problems or composite delaminations. CZM’s are based onspring (Cui and Wisnom 1993; Lammerant and Verpoest 1996) or more typicallycohesive elements (Mi et al. 1998; Petrossian and Wisnom 1998; Feraren and Jensen2004), connecting 2D or 3D solid elements of structures. An important feature ofCZM’s is that they can be easily incorporated in conventional FE softwares to modelthe fracture behaviour in various materials, including adhesively bonded joints (Jiet al. 2010). CZM’s are based on the assumption that one or multiple fractureinterfaces/regions can be artificially introduced in structures, in which damagegrowth is allowed by the introduction of a possible discontinuity in the displacementfield. The technique consists on the establishment of traction–separation laws(addressed as CZM laws) to model interfaces or finite regions.

The CZM laws are established between paired nodes of cohesive elements, andthey can be used to connect superimposed nodes of elements representing differentmaterials or different plies in composites, to simulate a zero thickness interface(local approach; Fig. 11a; Pardoen et al. 2005), or they can be applied directlybetween two non-contacting materials to simulate a thin strip of finite thicknessbetween them, e.g. to simulate an adhesive bond (continuum approach; Fig. 11b;Xie and Waas 2006).

Fig. 11 Cohesive elements to simulate zero thickness failure paths—local approach (a) and tomodel a thin adhesive bond between the adherends—continuum approach (b) in an adhesive bond


A few works on CZM techniques use the local approach (Campilho et al. 2005,Liljedahl et al. 2006, Turon et al., 2007a). With this methodology, the plasticdissipations of the adhesive bond are simulated by the solid finite elements, whilstthe applicability of cohesive elements is restricted to damage growth simulation(Fig. 11a). The CZM laws usually present an extremely high initial stiffness(penalty function method), not to change the structures global stiffness. Placementof the cohesive elements at different growth planes in the transverse direction ofthe adhesive bond is also feasible, capturing the respective stress gradients andconcentration towards the singular regions (Campilho et al. 2005). In the localapproach for bonded joints simulation, the adhesive is modelled as an elasto-plastic continuum by solid FE elements (Pardoen et al. 2005) and the ‘‘intrinsicfracture energy’’ is considered for the CZM laws instead of Gc, relating to therequired dissipation of energy to create a new surface, while the plastic dissipa-tions of ductile materials take place at the solid elements representative of theadhesive bond (Liljedahl et al. 2006). Thus, Gc is the sum of these two energycomponents, increasing by inclusion of the plastic dissipation of materials in themodels. Under these assumptions, damage growth is ruled by the work of sepa-ration of the fracture surfaces instead of Gc, due to the dissipated energy by thecontinuum elements. The effects of external and internal constraints on the plasticdissipations of an adhesive bond are thus accountable for in the local approach.On the other hand, compared to the continuum approach, to be described in thefollowing, more parameters and computations are needed (Ji et al. 2010).

CZM’s have also been used to simulate the behaviour of adhesive bonds by acontinuum approach (Fig. 11b), by the replacement of the entire adhesive bond bya single row of cohesive elements with the representative behaviour of theadhesive bond (Kafkalidis and Thouless 2002; Campilho et al. 2008a). The initialstiffness of the cohesive elements, unlike for the local approach, represents theadhesive bond stiffness in each mode of loading, and the global behaviours of theadherends are fully correlated by these elements. Due to the evident simplicity ofthis approach, it has been widely used in the damage growth simulation of bondedjoints, giving accurate results providing that accurate calibrations are undertakenfor the CZM laws (Campilho et al. 2009a). Despite the computationally efficiencyof continuum CZM modelling of bonded joints, few limitations exist: (1) thephysical meaning of the fracture process was somehow lost, because real cohesiveseparations are usually accompanied with localized plastic behaviours across theadhesive interlayer, even for brittle adhesives, represented with this method byaveraged equivalent properties and (2) CZM’s become dependent of the structuregeometry, more specifically on tP and tA, because these largely affect the size ofthe fracture process zone (FPZ) and plasticity around the crack tip, thus making theCZM laws dependent of these parameters (Ji et al. 2010).

From this discussion, it is evident that CZM’s provide a macroscopic repro-duction of damage along a given path, disregarding the microscopic phenomena onthe origin of failure, by the establishment of a traction-relative displacement (t – d)response between paired nodes on opposite faces of the pre-defined crack path, byspecification of large scale parameters ruling the crack growth process such as Gc

n,


Gcs or Gc

t (Zhao et al. 2011b). CZM’s thus simulate strength evolution and sub-sequent softening up to complete failure, to account for the gradual degradation ofmaterial properties. The traction–separation laws are typically represented bylinear relations at each one of the loading stages (Yang and Thouless 2001),although one or more stages can be defined differently for a more accurate rep-resentation of the materials behaviour. Figure 12 represents the 2D triangularCZM model actually implemented in Abaqus� for static damage growth, which isdescribed here in detail. The 3D version is similar, but it also includes the tearingcomponent. More details on the 3D CZM implemented in Abaqus� can be foundin the work of Campilho et al. (2011b) or in Abaqus� Documentation (2009).

Regarding the nomenclature used throughout this work, the subscripts n ands relate to pure normal (tension) and shear behaviours, respectively. tn and ts arethe corresponding current stresses, and dn and ds the current values of d. CZM’srequire the knowledge of Gn and Gs along the fracture paths and respective valuesof Gc

n and Gcs . Additionally, the cohesive strengths must be defined (tn

0 for tensionand ts

0 for shear), relating to the onset of damage, i.e., cancelling of the elasticbehaviour and initiation of stress softening. dn

0 and ds0 are the peak strength dis-

placements, and dnf and ds

f the failure displacements. The values of dnf and ds

f aredefined by Gn

c or Gsc, respectively, as these represent the area under the CZM laws.

As for the mixed mode CZM law (Fig. 12), tm0 is the mixed mode cohesive

strength, dm0 the corresponding displacement, and dm

f the mixed mode failuredisplacement.

Under pure mode loading, the respective t-d response attains its peak at thecohesive strength (tn

0 or ts0), corresponding to damage initiation by the induced

reduction of stiffness of the cohesive element. Softening follows and, when thevalues of t are completely cancelled, the crack propagates up to the adjacent set ofpaired nodes in the failure path, permitting the gradual debonding between crackfaces. Under mixed loading (i.e., when two or three modes of loading aresimultaneously present), stress and/or energetic criteria are often used to combinethe pure mode laws, thus simulating the typical mixed mode behaviour inherent tobonded assemblies. By the mentioned principles, the complete failure response ofstructures can be simulated (Zhu et al. 2009). For the estimation of the cohesivelaw parameters, a few data reduction techniques, described further in detail, areavailable (e.g. the property determination technique, the direct method and the

Fig. 12 Triangular CZM law(adapted from Abaqus�

Documentation 2009)


inverse method) that enclose varying degrees of complexity and expected accuracyof the results (Sørensen and Jacobsen 2003; Li et al. 2005a; Liljedahl et al. 2006).

A CZM thus extends the concepts of continuum mechanics described inSect. 2.2 by including a zone of discontinuity when the pre-defined CZM path isfulfilled, using characteristic parameters of the continuum mechanics approachesfor the onset of damage and energy parameters of fracture approaches for prop-agation (Banea and da Silva 2009; Chen et al. 2011b). This allows CZM’s to be ofmuch more general utility than conventional fracture mechanics, because of thesimulation of onset and non-self-similar growth of damage without user inter-vention, and non-dependence on an initial flaw, unlike conventional fracturemechanics approaches. Compared to continuum mechanics approaches, CZM’sare mesh independent provided that sufficient integration points are simultaneouslyunder softening at the crack front (Campilho et al. 2009b; Campilho et al. 2011a).Actually, when the mixed mode softening initiates at a given set of paired nodes,material degradation initiates, leading to a stress reduction and consequent stressredistribution to the neighbouring sets of nodes. If this is performed smoothly bysufficiently small load increments, then a stable propagation takes place, avoidingany mesh influence. CZM’s have been largely used in recent years to simulate thebehaviour of structures up to failure, as they allow to include in the numericalmodels multiple failure possibilities, within different bulk regions of materials orbetween materials interfaces, e.g. at the adhesive bonding interfaces. Theknowledge of the loci of damage occurring in a structure is not required as an inputparameter, as the CZM globally searches damage initiation sites at specified failurepaths satisfying the established criteria. However, cohesive elements must exist atthe planes where damage is prone to occur, which, in several applications, can bedifficult to know in advance. Although a large amount of CZM failure paths maybe introduced in a structure to simulated different fractures (Campilho et al.2009a), it is not feasible to introduce cohesive elements between every field ele-ment, even in a moderate size mesh. However, an important feature in bondedassemblies is that damage propagation is restricted to well defined planes, i.e., at ornear the adhesive/adherend interfaces between the adhesive and the adherends, orcohesively in the adhesive bond, which allows overcoming this limitation(Campilho et al. 2008a). Several CZM law shapes have been presented in theliterature, depending on the nature of the material or interface to be simulated. Thetriangular, exponential and trapezoidal shapes, as shown in Fig. 13, are the mostcommonly used for strength prediction of typical materials. For the representedtrapezoidal CZM, dn

s and dss are the stress softening onset displacements.

Selection of the Cohesive Law Shape

The shape of the CZM laws can be adjusted to conform to the behaviour of the thinmaterial strip or interface they are simulating. Developed models over the yearsinclude triangular (Alfano and Crisfield 2001), linear-parabolic (Allix andCorigliano 1996), polynomial (Chen 2002), exponential (Chandra et al. 2002) and


trapezoidal laws (Kafkalidis and Thouless 2002). Thus, CZM’s can be adapted tosimulate ductile adhesive bonds, which can be modelled with trapezoidal laws(Campilho et al. 2008a, 2009a). It was recently made possible to define the preciseCZM shape by advanced techniques, although this is always a difficult task to beperformed by experimental procedures. As a result, the CZM is often assumed orsimplified based on the perceived knowledge of the materials/interfaces to besimulated (Kafkalidis and Thouless 2002). Notwithstanding this fact, varyingpublished results gave predictions within acceptable errors even for small varia-tions to the optimal CZM parameters and shapes (Liljedahl et al. 2006).

On the other hand, the effect of the CZM law shape significantly variesdepending on the structure geometry and behaviour of the materials. These issuesbecame evident in the experimental and FE study of Pinto et al. (2009), whichpredicted the tensile strength of single lap joints between adherends with similarand dissimilar materials and values of tP, bonded with the adhesive 3 MDP-8005�. The authors concluded that for stiffer adherends, the precise shape ofthe cohesive law is fundamental for the accuracy of the results and load–displacement (P–d) response of the structure, since peel loads are minimized andthe joint is mainly loaded by shear. Due to the large longitudinal stiffness, dif-ferential deformation effects are minimized as well, which results on identicalshear loading along the adhesive bond. As a result, the P–d curve precisely followsthe shape of the CZM law. Compliant adherends give rise to substantial shear andpeel peak stress gradients along the adhesive bond, diluting the miscalculation ofcurrent stresses over the entire overlap. The results of Ridha et al. (2010) alsoemphasized on the CZM shape influence, by modelling a thin adhesive bond of ahigh elongation epoxy adhesive FM� 300 M (Cytec) in scarf repairs on compositepanels. Linear, exponential and trapezoidal softening was considered to modelplasticity of the adhesive, with the use of linear softening resulting in underpredictions of the repairs strength by nearly 20%, because of premature softeningat the bond edges after tm

0 (Fig. 12) that is not consistent with the actual behaviourof the adhesive.

In which regards to the suitability of the CZM laws for adhesive bonds, atrapezoidal law is often preferred for ductile adhesives (Feraren and Jensen 2004;

Fig. 13 Different shapes of pure mode CZM laws: triangular or exponential (a) andtrapezoidal (b)


Campilho et al. 2010), especially when extremely stiff components are bonded,due to the practically absence of differential deformation effects in these compo-nents (Alfano 2006, Pinto et al. 2009). A triangular CZM is commonly selected forbrittle materials that do not undergo plasticization after yielding (Campilho et al.2011a), and also for the intralaminar fracture of composite adherends in bondedstructures, due to their intrinsic brittleness (Xie et al. 2006).

For the selection of the most appropriate CZM law shape for a given appli-cation, the material/interfacial behaviour should always be the leading decisionfactor. Nonetheless, the CZM law shape also influences the iterative solvingprocedure and the time required to attain the solution of a given engineeringproblem. Actually, larger convergence difficulties are expected for trapezoidalrather than triangular CZM laws, mainly because of a more abrupt change in thecohesive elements stiffness during stress softening. Actually, for a specified valueof Gn

c or Gsc, the larger the value of dn

s or dss of the trapezoidal law (Fig. 13b), the

larger is the variation to the triangular descending slope. In summary, the selectionof CZM law cannot be separated from any of these issues.

Triangular Cohesive Zone Model

The triangular CZM law, described in detail in this section, is the most commonlyused due to its simplicity, reduced number of parameters to be determined, andgenerally acceptable results for most real conditions (Liljedahl et al. 2006).However, generically speaking, the shape of the cohesive laws can be adjusted toconform to the behaviour of the material or interface they are simulating for moreprecise results (Campilho et al. 2009a). The current section describes the statictriangular 2D CZM implemented in Abaqus� v6.10 (Fig. 12). Different shapeCZM’s are based on this formulation, typically differing on the softeningsimulation.

Fatigue CZM’s are also invariably based on the described formulation in thissection but, as no software implementation is yet available and different approa-ches have been proposed, these are characterized in ‘‘Fatigue Applications ofCohesive Zone Models’’, which specifies fatigue CZM’s in detail. Under puremode, damage propagation occurs at a specific set of paired nodes when the valuesof tn or ts are released in the respective CZM law. Under mixed mode, stress andenergetic criteria are often used to combine tension and shear (Campilho et al.2008a).

Cohesive elements are assumed to be under one direct component of strain(tension) and one transverse shear strain, which are computed directly from theelement kinematics. The membrane strains are assumed as zero, which is appro-priate for thin and compliant bonds between stiff adherends. Undamaged strengthevolution is defined by a constitutive matrix relating the current stresses and strainsin tension and shear across the cohesive elements (subscripts n and s, respectively)(Abaqus� Documentation 2009)


t ¼ tn

ts

� �¼ Knn Kns

Kns Kss

� �:

en

es

� �¼ KCOHe: ð5Þ

The matrix KCOH contains the stiffness parameters of the adhesive bond, whosedefinition depends on whether the local or continuum approach is being used. Inthe local approach, used to simulate zero thickness fractures, the KCOH parametersare chosen as an extremely large value (penalty function method) for the cohesiveelements not to interfere with the structure deformations, given that this task is tobe performed solely by the continuum FE elements (Campilho et al. 2005). Forcontinuum CZM modelling of bulk thin strips and more specifically for adhesivebonds, a suitable approximation is provided with Knn = E, Kss = G, Kns = 0(Abaqus� Documentation 2009); E and G are the longitudinal and transverseelastic moduli, respective. Under these conditions, the model response for theadhesive bond accurately reproduces its deformation behaviour. A few userimplemented models in the literature (Campilho et al. 2008a) specify Eq. 5directly in terms of t–d relationship; d representing the vector of relative dis-placements including dn and ds. With this formulation, Knn and Kss are defined asthe ratio between E or G and tA.

Damage initiation under mixed mode conditions can be specified by differentcriteria. In this description, the quadratic nominal stress criterion for the initiationof damage is presented, previously tested for accuracy (Campilho et al. 2009a),expressed as (Abaqus� Documentation, 2009)

tnh it0n

� �2

þ ts

t0s

� �2

¼ 1: ð6Þ

h i are the Macaulay brackets, emphasizing that a purely compressive stressstate does not initiate damage (Jing et al. 2009). After tm

0 is attained (Fig. 12) bythe fulfilment of Eq. 6, the material stiffness initiates a softening process. This issimulated by the energy being released in a cohesive zone ahead of the crack tip(FPZ). This region is where the material undergoes damage by different ways(Pereira and de Morais 2003), e.g. micro-cracking, extensive plasticity and fibrebridging (e.g. for composite adherends). Numerically, this is implemented by adamage parameter whose values vary from zero (undamaged) to unity (completeloss of stiffness) as the material deteriorates. Complete separation (dm

f in Fig. 12)are predicted by a linear power law form of the required energies for failure in thepure modes (Abaqus� Documentation 2009)

Gn

Gcn

þ Gs

Gcs

¼ 1: ð7Þ


Convergence Issues

The use of CZM’s for static or fatigue strength prediction is intrinsically linked toconvergence issues that may occur in the incremental iterative process in whichthe load step is divided into, at the time of reaching to the solution of the system ofequations expressed in Eq. 1. As previously specified in Sect. 2.1, in a givenincrement, solution is attained when the iterative procedure gives an error betweenthe left and right sides of the fundamental equation under a software pre-defined orused-defined value (Abaqus� Documentation 2009).

Usually, the main factors contributing to convergence difficulties are thepropagation of damage by CZM formulations (which depend on the CZM lawshape) and respective instability, the consideration of plastic constitutive behav-iours for the solid elements, the adhesive plasticity, the adherends stiffness, theorthotropic nature of composite materials, coarse meshes at or near stress con-centration/damage propagation sites and structural geometries with large stressconcentrations, singular regions or intersecting cracks (Hamitouche et al. 2008),more specifically if two or more of these features exist concurrently within ananalysis. In FE models, the increments usually run smoothly up to damage uptakeby the cohesive elements along the failure paths, after which convergence prob-lems appear due to the abrupt variation of K (Eq. 1) between consecutiveincrements.

Obviously that, as it was previously mentioned, trapezoidal laws give moreconvergence problems because of the more abrupt change in the cohesive elementsstiffness during softening. Gradual softening and reduction of the mesh size at thefracture regions can or not be enough to surpass difficulties and attain total sep-aration. Convergence problems may also occur during unstable crack propagation,when the available value of G is higher than the respective Gc (Abaqus� Docu-mentation 2009), which can be prevented by smaller step incrementations.Isotropic plasticity in the adherends or orthotropic elasticity for composite adh-erends creates larger difficulties during CZM damage growth and, under theseconditions, elastic-perfectly plastic approximations or approximation to isotropicequivalent properties, respectively, can assist to a smoother fracture withoutcompromising the results. On the same line of the adherends plasticity, compliantadherends such as polymers usually endure significant deformations and elementdistortions near the damage propagation sites, with the respective difficulty toreduce the iterative errors during crack growth for implicit solving techniques suchas in Abaqus�/Standard. There is not an ultimate solution to improve convergencefor this scenario, although the local reduction of the mesh size or use of geometricnon-linearities in the FE software may help. Concerning the adhesive behaviour,brittle adhesives turn the CZM softening process more abrupt, notwithstanding theCZM law shape that, added to the smaller FPZ’s at the crack tip, result in largerconvergence difficulties. Depending on the bonded system, the increase of Gn

c

and/or Gsc is sometimes enough to achieve surface separation, although a small

over prediction of the structural strength is naturally associated (up to 10% insingle lap joints for 100% increase of Gn

c and Gsc; Campilho et al. 2011c). All of the


other issues, such as coarse meshing, singularities or intersecting cracks are par-tially solved by a bigger mesh refinement.

Transversely to all of these scenarios, the reduction of the cohesive elementsinitial stiffness up to tn

0 or tn0 is equally a possibility for the reduction of conver-

gence problems by the reduction of the adjacent elements distortion and stressconcentrations. However, this must be performed in such a manner that thestructure’s overall stiffness is not compromised (for both local and continuumapproaches) and, additionally for adhesive bonds modelled by a continuum CZM,bearing in mind that the constitutive behaviour of the bond is being altered.Interpenetration of the crack faces can also occur for small stiffness values andshould be prevented as well (Khoramishad et al. 2010). Following extensivestudies, the value of 106 N/mm3 was recommended by Gonçalves et al. (2000) fora triangular static CZM (local approach modelling) for an accurate convergenceand minimization of convergence difficulties during the nonlinear procedure.Nonetheless, values down to 59104 N/mm3 were used without compromising theglobal behaviour of bonded repairs on composite structures (Campilho et al.2008b). Smaller values are not recommended on account of the stiffness accuracyissues and numerical ill-conditioning problems (Khoramishad et al. 2010).

Apart from the proposed solutions, FE software packages usually hold a fewtechniques to aid the iterative process. Abaqus�, as an example, gives the userthree main possibilities (Abaqus� Documentation 2009): (1) viscous regulariza-tion, (2) automatic stabilization and (3) non-default solution controls.

Using viscous regularization in Abaqus�/Standard improves the convergence offracture problems, by implementation of viscous regularization within the con-stitutive equations, which allows stresses to be outside the limits of the CZM laws.A small value for the viscosity parameter, compared to the characteristic timeincrement, usually improves the convergence rate of simulations during the soft-ening regime, without compromising the accuracy of the results.

Consideration of automatic stabilization is another possibility to improveconvergence for unstable problems (e.g., due to buckling or material softening). Ifthe instability is localized, a local transfer of strain energy takes place from onepart of the model to the neighbouring regions, and global solution methods maynot work (Abaqus� Documentation 2009). These difficulties have to be solvedeither dynamically or with the introduction of (artificial) damping. Abaqus�/Standard uses an automatic algorithm for the stabilization of locally unstablequasi-static problems by the automatic addition of volume-proportional dampingto the model.

The use of non-default solution controls consists on the user modification of thenonlinear equation solution accuracy and time increment adjustment, to activelyaffect the incremental solving procedure and solver parameters to force conver-gence. However, caution is required, since if less strict convergence criteria areimposed, results may be accepted as converged although they are not sufficientlyclose to the exact system solution. The line search algorithm is also withinthe scope of solution controls, allowing the detection of divergent equilibriumiterations in strongly nonlinear problems solved by the Newton algorithms of


Abaqus�/Standard. A scale factor is then applied to the computed solution cor-rection, which helps to prevent divergence.

The work of Khoramishad et al. (2010), relating to the development of a fatigueCZM for single lap joints, included a preliminary study on the viscous regulari-zation technique of Abaqus� as a requirement for the attainment of static failure,for further progress to the fatigue damage implementation. The authors empha-sized on the necessity of not changing the overall structural response by theinclusion of viscous damping in the models. As a result, the optimal value of thisparameter was found by a parametric study with decreasing levels of implementedviscous damping coefficient, l (Fig. 14).

Figure 14 shows the result of a non-optimized value of l (10-2 N.s/mm), as theresponse of the bonded joints are largely affected, and the final optimized resultsby consideration of l = 10-5 N.s/mm, since this quantity allowed surpassing thedivergent behaviour of the solution although it did not change the overall results.

Determination of the Cohesive Parameters

CZM analyses offer a powerful means to account for the largely nonlinear fracturebehaviour of modern adhesively bonded joints, but the CZM parameters requirecareful calibrations by experimental data and respective validation in order toaccurately simulate the failure process (DuBois 2004; Carlberger and Stigh 2010).Despite this fact, standardized methods for the definition of tn

0 and ts0 are not yet

available (Lee et al. 2010). In recent years, many works were published regardingthe definition of the CZM parameters (Gn

c, Gsc, tn

0 and ts0) and a few data reduction

techniques are currently available (e.g. the property determination technique, thedirect method and the inverse method) that enclose varying degrees of complexityand expected accuracy of the results. The few of these works that validated withmixed mode experiments the estimated pure mode CZM’s typically made use of

Fig. 14 The effect of l onthe predicted P–d curve of asingle lap joint (Khoramishadet al. 2010)


double-cantilever beam (DCB), end-notched flexure (ENF) or single lap speci-mens, generally with good results (Kafkalidis and Thouless 2002; Zhu et al. 2009).

The property determination method, consisting on the isolated definition of allthe cohesive law parameters by suitable tests, is particularly critical if bulk testsare used, owing to reported deviations between the bulk and thin adhesive bondcohesive properties (Andersson and Stigh 2004). This is caused by the strainconstraining effect of the adherends in bonded assemblies, and also by the typicalmixed mode crack propagation in adhesive bonds (Andersson and Stigh 2004;Högberg and Stigh 2006; Högberg et al. 2007; Leffler et al. 2007). Actually, inbulk materials cracks tend to grow perpendicularly to the direction of maximumprincipal stress (Chai 1992). In thin bonds, cracks are forced to follow the bondlength path since, as the adhesive is typically weaker and more compliant than thecomponents to be joined, failure is often cohesive within the adhesive (Zou et al.2004; Campilho 2005). The inverse method consists on the trial and error fittinganalysis to experimental data on bonded joints, such as the P–d curve, allowingtuning of the simplified shape CZM laws for particular conditions. For the propertydetermination technique and the inverse method, a simplified parameterized shapeis often selected (e.g. bilinear or trilinear) for the CZM, based on an assumptionmade by the user depending on the material behaviour to be simulated (Campilhoet al. 2009a). On the other hand, in the direct method the precise shape of the CZMlaws is easily defined, since it computes the CZM laws of an adhesive bond fromthe measured data of fracture characterization tests (Pandya and Williams 2000) bydifferentiation of the Gn–dn or Gs–ds curves. However, for subsequent use by FEstrength prediction techniques, it is common practice to build a simplifiedapproximation for easier implementation.

Notwithstanding the parameter identification method, deviations are expectedto occur between the quantitative prediction of the cohesive parameters and thereal behaviour of the adhesive bond (Leffler et al. 2007; Pinto et al. 2009).Contrarily to the property determination method, the inverse and direct methodsprovide more accurate estimations as the adhesive can be characterized underidentical adherend restraining conditions to real applications, giving accurateestimations (Pardoen et al. 2005). However, the direct method is still considered togive the most accurate results, since it provides the precise shape of the CZM laws,while the inverse method parameters are based on previously assumed CZMshapes. Concerning the CZM parameters, Gc is usually the key parameter to bedetermined, because of the role that it plays on the overall results. For its esti-mation, the LEFM-based methods are usually easier to apply, although these canonly be used for brittle or moderately ductile adhesives. For adhesives that endureextensive plasticization, LEFM techniques are rendered unfeasible, and theJ-integral method emerges as a viable alternative. However, it is more complicatedto apply, as generally additional data is required from the tests, such as the adh-erends rotation during loading.

Disregarding the parameter determination method, the CZM parametersinvariably depend on tA and tP, which emphasizes the importance of the tA and tPconsistency between the fracture tests and the structures to be simulated (Bascom


and Cottington 1976; Chai 1986a, b; Andersson and Stigh 2004; Leffleret al. 2007).

Fracture Characterization Methods

The values of Gnc and Gs

c are estimated by standardized test methods for all of thethree data reduction methods. Additionally, the inverse and direct methods, alsotake advantage of the P–d results of the fracture tests for the accurate derivation ofthe CZM laws in each mode of loading. The DCB test for tension (Fig. 15a) andthe ENF test for shear (Fig. 15b) are the most widespread test methods (B is thespecimen width, a0 the initial crack length and L the specimen length; DCB test, orhalf-length; ENF test).

Several works are available for fracture characterization in tension by the DCBtest (Bader et al. 2000; Ducept et al. 2000; Nairn 2000; Andersson and Stigh 2004;Hojo et al. 2006). The main advantages of this test method include its simplicityand the possibility to obtain Gn

c mathematically using the beam theory (Yoshihara2007). However, some issues must be taken into account for an accurate estima-tion. In fact, unstable crack propagation was experimentally detected by Baderet al. (2000) and Ducept et al. (2000), which hinders a clear crack length(a) monitoring during the test. In other cases, in the DCB test of adhesively bondedstructures, the crack tip may not be clearly visible depending on the adhesive. Thiscan induce non-negligible errors on the derivative of the compliance (C = d/P) relatively to a used in the compliance calibration method (CCM). On the otherhand, the energy dissipated at the FPZ can be large for ductile adhesives, implyingthat beam theory-based methods without any corrections will underestimate Gn

c.A few methods are available that allow the direct extraction of Gn

c, accounting forthe FPZ effects and not requiring the crack length measurement during propagation

Fig. 15 DCB (a) and ENF(b) tests for fracturecharacterization of thinadhesive bonds


(Tamuzs et al. 2003; Biel and Stigh 2008; de Moura et al. 2008). Blackman et al.(2003) considered tapered double-cantilever beam (TDCB) and peel tests insteadto assess Gn

c of an adhesive layer and to study fracture of adhesively bonded joints.Fracture characterization in shear is still not well addressed owing to some

inherent features of the most popular tests: ENF, end-loaded split (ELS) and four-point end-notched flexure (4ENF). Actually, the ELS test involves clamping one ofthe specimen edges, constituting a source of variability and increasing the com-plexity of data reduction (Blackman et al. 2005). On the other hand, the 4ENF testrequires a complex setup and presents some problems related to large frictioneffects (Schuecker and Davidson 2000). Therefore, the ENF test is the most suitedfor shear fracture characterization of adhesively bonded structures (Leffler et al.2007). However, problems related to unstable crack growth and to crack moni-toring during propagation are yet not fully solved. In addition, the classical datareduction schemes, based on beam theory analysis and compliance calibration,require the monitoring of a during propagation. Added to these issues, a quiteextensive FPZ develops ahead of the crack tip for ductile adhesives, which affectsthe measurement of Gs

c. Consequently, its influence should be taken into account,which does not occur when the real value of a is used in the selected data reductionscheme. To overcome these limitations, data reduction schemes for the measure-ment of Gs

c that are only based on the specimens compliance are also available(de Moura et al. 2009).

An important issue to account for when using the direct or inverse methods forthe estimation of the CZM laws is to match as possible the values of tP, lay-up (forstacked composite laminates) and tA between the fracture tests and the bondedstructures to be simulated (Campilho 2009), since the cohesive parameters of theadhesive bond largely depend on the adherends restraining to the adhesive bond.Pardoen et al. (2005) performed a parametric study of the effect of tP on the valuesof Gn

c, using TDCB and compact tension (CT) fracture tests. It was shown that theincreased bending of thin adherends promoted plasticity in the adhesive bond,greatly increasing Gn

c. Actually, the large transverse deflection of thin adherends isassociated with large root rotation and substantial shear stresses, which tend topromote plastic yielding in the adhesive, increasing plastic strains as well.Additionally, extra plastic dissipation associated with the bending of the adhesivebond develops around the crack tip.

Particular attention should also be paid to match tA between the fracturecharacterization tests and the real structures to be simulated, as previously men-tioned in ‘‘Determination of the Cohesive Parameters’’. In fact, adhesives in theform of thin bonds behave differently than as a bulk material (Andersson and Stigh2004), because of the strain constraining effects of the adherends and therespective typical mixed mode crack propagation (Högberg and Stigh 2006, Leffleret al. 2007). Different studies reported on an obvious dependence of Gc of adhesivebonds with tA (Bascom et al. 1975; Bascom and Cottingham 1976; Hunston et al.1989; Bell and Kinloch 1997; Lee et al. 2003; Lee et al. 2004). Typically, the valueof Gc of a thin adhesive bond increases with tA up to a peak value, bigger than thebulk quantity (Kinloch 1987; Chai 1988; Crews et al. 1988). After, Gc decreases


with tA to a steady-state value, corresponding to Gc of the bulk adhesive (Ikedaet al. 2000; Duan et al. 2002, 2003). This tendency is consistent with the works ofYan et al. (2001a, b), which studied the influence of tA on the fracture properties ofDCB and CT joints with aluminium adherends and a rubber-modified epoxyadhesive. Using a large deformation FE technique and the peak loads measured inthe experiments, the critical value of the J-integral was calculated for differentvalues of tA. Biel (2005) also emphasized on the bigger values of Gc near theoptimal value of tA than as a bulk. This was due to the predominantly state ofprescribed deformation at the region where the crack could propagate, whichenlarged the FPZ. Actually, with tough engineering adhesives, near the optimalvalue of tA the FPZ typically extends several times larger than the value of tA, andsubstantially longer than in bulk adhesives, resulting on bigger values of Gc.

Property Determination Method

In the property determination method, at least one of the CZM law parameters isapproximated by consideration of bulk adhesive properties (Pinto et al. 2009).Campilho et al. (2008a) evaluated the tensile strength of bonded single straprepairs on laminated composites as a function of the overlap length and the patchthickness. A FE analysis including a trapezoidal CZM was used to simulate a thinductile adhesive bond of Araldite� 420 (Hunstman) by the continuum approach. Inthe authors work, tn

0 and dns were obtained from the r–e curve of the bulk adhesive,

substantiated by previous evidence (Andersson and Stigh 2004) that tn0 is of the

same order of magnitude of the tensile strength measured in bulk tests, and that dn0

and dns do not significantly influence the numerical results (Yang et al. 1999). On

the other hand, ts0 was derived from tn

0 by the von Mises yield criterion for bulkisotropic materials and, owing to its less influence on the results (Yang et al. 1999),dn

s was defined considering a similar softening slope to the tension CZM law(Carlberger and Stigh 2007). The values of Gn

c and Gsc were estimated from DCB

and ENF tests, respectively. The interlaminar, intralaminar and fibre failureproperties of the composite were obtained from previous works, considering tri-angular CZM laws (local approach). Different fracture paths were equated(Fig. 16; A–A represents the line of symmetry for the FE simulations), in order toaccount for different failure modes. A reasonable agreement was found for the

Fig. 16 Location of theCZM elements in the singlestrap repairs (Campilho et al.2008a)


stiffness and failure load/displacement, despite the aforementioned CZM param-eter approximations.

Nonetheless, for an evaluation of the effect of the described approximations onthe predictions, a sensitivity analysis was also performed by the authors, showingthat dn

s and dss of the adhesive bond CZM laws do not change the failure mode of

the repairs nor do they affect the failure load. Accounting for the observed failures,at lines P1 and P3 (interlaminar failure of the composite), the shear interlaminarCZM parameters showed a larger influence on the strength than the tension ones.

Pinto et al. (2009) tested a trapezoidal CZM with a continuum formulation tomodel a tA = 0.2 mm adhesive bond (3 M DP-8005�) for the estimation of thetensile strength of single lap joints between adherends with different values of tAand materials (polyethylene, polypropylene, glass-epoxy and carbon-epoxy com-posites). The FE analysis was carried out in Abaqus�, with user developed CZMelements implemented in a sub routine. For the tensile CZM law, tn

0 and dns

(Fig. 13b) were assumed to be equal to the corresponding bulk quantities.Although the authors did emphasize on possible miscalculations arising from thisprocedure, this course of action was substantiated by previous results (Anderssonand Stigh 2004). Gn

c was estimated from DCB tests using the ASTM D3433-99(2005) standard. The value of dn

s was approximated by the product of the averagefailure strain obtained in the adhesive bulk tests with tA, as this parameter does notsignificantly influence the FE results (Yang et al. 1999).

Oppositely to this course of action, the CZM parameters in shear loading wereobtained with an inverse method (the basic principles of this technique aredescribed in detail in ‘‘Inverse Method’’), considering the block-shear test method(ASTM D4501-01 (2009). The block shear test was adopted since the adhesivelayer is mainly loaded in shear, while normal stresses are minimized. The FEsimulations captured fairly accurately the experimental behaviour of the joints, interms of stiffness, and maximum load and the corresponding value of d for all ofthe geometry and material configurations tested.

Direct Method

By the direct method, the complete CZM law and respective shape for a givenmaterial strip or interface can be precisely estimated by the differentiation of theGn–dn or Gs–ds curves. A few works currently exist on the direct parameterdetermination of adhesive bonds (Sørensen 2002; Zhu et al. 2009) and fiber-reinforced composites (Sørensen and Jacobsen 2003). Andersson and Stigh (2004)used a direct method to determine the continuum CZM parameters in tension of aductile adhesive bond of Dow Betamate� XW1044-3 in a DCB test configuration,after approximation of the Gn–dn data to a series of exponential functions to reduceerrors in the measured data. The results showed that the tn–dn relationship can bedivided in three parts. Initially tn increases proportionally to dn (linear elasticbehaviour of the adhesive), until a limit stress is achieved. A plateau regionfollowed, corresponding to the development of plasticity in the adhesive.


The curve ended with a parabolic softening region, giving an approximate trap-ezoidal shape.

Högberg and Stigh (2006) proposed a direct integration scheme to capture theCZM laws (continuum approach) of an adhesive bond by a mixed-mode DCBspecimen, which consisted on the differentiation of the J-integral vs. dn (tensionCZM law) or ds (shear CZM law). The J-integral was derived from the test data bya developed formulation that required the adherends rotation at the loading point,h. Results showed that substantial differences exist between pure tensile and shearCZM laws, and that ten evenly distributed mode mixities can catch the mixedmode constitutive behaviour of an adhesive system for any load and geometryconditions.

In the work of Carlberger and Stigh (2010), the continuum CZM laws of a thinbond of a ductile adhesive (Dow Betamate� XW1044-3) were determined intension and shear using the DCB and ENF test configurations, respectively, con-sidering 0.1 B tA B 1.6 mm. The values of Gn and Gs were derived by a J-integralformulation to accurately capture the large plastic straining effects present at thecrack tip of the ductile adhesive (Yan et al. 2001a, b). Actually, the J-integral is aviable means to capture the adhesive nonlinearity for a monotonic loading process,i.e., if no unloading occurs. The cohesive laws were derived by a direct methodthat used a least square adaption of a Prony-series to the Gn/Gs versus dn/ds data, toavoid errors on the measured data resulting from direct differentiation of theexperimental results. Although the J-integral solutions for this purpose have itsadvantages, the respective formulae require additional data to the P–d dataavailable from the tests than LEFM techniques. Actually, on one hand, for Gn

determination by the DCB test method, the applied force, P, and h are necessaryparameters to be defined as functions of dn. On the other hand, for shear charac-terization by the ENF test method, the simultaneous measurement of the appliedforce, P, and shear sliding at the tip of the adhesive bond, ds, are required. Withthis data, by direct application of formulae derived in the authors work, the evo-lution of Gn and Gs (in the authors work addressed as J) with the deformation ofthe adhesive bond could be established (Fig. 17).

Fig. 17 Gn–dn and Gs–ds

relations for the adhesiveDow Betamate� XW1044-3and tA = 0.2 mm (Carlbergerand Stigh 2010)


Differentiation of the Prony-series adaptions of Gn or Gs relatively to the rel-ative displacements, dn or ds, gives the full cohesive law up to failure, providedthat the curves of Fig. 17 are obtained up to a steady-state value (Zhu et al. 2009).

Figure 18 shows the averaged CZM laws in tension for each value of tA,considering 5–8 specimens for each condition. The obtained results clearly showthe transition between approximate triangular CZM laws for small values of tA totrapezoidal laws for bigger values. This is obviously related to an increase of Gn

c,because of larger FPZ developments in the adhesive bond (Choi et al. 2004; Duanet al. 2004). Figure 19 represents the shear CZM laws for tA values between 0.1and 1.0 mm, averaged over at least two specimens. The reported results arequalitatively consistent with the tension data, i.e., a CZM shape dependence withtA, with an approximate triangular shape for tA = 0.1 mm and modification to atrapezoidal shape for bigger values of tA. However, above tA = 0.2 mm the valueof Gs

c is virtually unaffected. The value of ts0 slightly diminished with tA because of

minor reductions of the effective strain rate with increasing values of tA(Carlberger et al. 2009).

It was thus concluded that the CZM shapes and respective parameters signifi-cantly vary with tA, ranging from a rough triangular shape for the smaller values oftA to a trapezoidal shape for bigger values of tA.

The continuum CZM analysis of Ji et al. (2010) addressed the influence of tA ontn0 and Gn

c for a brittle epoxy adhesive (Loctite� Hysol 9460), by using the DCBspecimen and the direct method for parameter identification. The analysis meth-odology relied on the measurement of Gn

c by the analytical J-integral methodproposed by Andersson and Stigh (2004), requiring the measurement of h.Figure 20 shows the testing setup for the measurement of Gn.

The testing machine was set to collect the load, load–point displacement (withlinear transducers placed at the crack tip) and, for the measurement of h, twodigital inclinometers with a 0.018 precision were attached at the free end of eachadherend. A charge-coupled device (CCD) camera with a resolution of3.7 9 3.7 lm/pixel was also used during the experiments, adjusted perpendicu-larly to one of the DCB specimen side edges. The camera focused at the crack tipand it collected images, which were input in an image processing toolkit for theestimation of dn at the crack tip, necessary for correlation with the load and h for

Fig. 18 CZM laws intension for the adhesive DowBetamate� XW1044-3 and0.1 B tA B 1.6 mm(Carlberger and Stigh 2010)


the definition of Gn. The tn–dn cohesive laws were quickly obtained by differen-tiation of the Gn–dn data (Sørensen 2002; Zou et al. 2009). Figure 21 reports on atypical Gn–dn relationship for tA = 0.2 mm and respective 6th degree polynomialfunction for differentiation and respective estimation of the CZM law.

Representative CZM laws for the adhesive layer are depicted in Fig. 22,showing some interesting variations, such as the previously mentioned increase ofGn

c with tA, perceptible by the increase of area under the tn–dn curve (Carlbergerand Stigh 2010). A reduction of tn

0 was also found with bigger values of tA. Another

Fig. 19 CZM laws in shearfor the adhesive DowBetamate� XW1044-3 and0.1 B tA B 1.0 mm(Carlberger and Stigh 2010)

Fig. 20 DCB specimen inthe testing machine withinclinometers attached to theadherends and camera formeasurement of h and dn,respectively (Ji et al. 2010)

Fig. 21 Relationshipbetween Gn and dn fortA = 0.2 mm (Ji et al. 2010)


important finding was related to the value of tn0 = 88 MPa found for

tA = 0.09 mm, compared to the bulk strength of 30.3 MPa. However, forincreasing values of tA the estimated tn

0 values quickly approached the bulkstrength of the adhesive (Fig. 22). The authors concluded that the J-integralmethod for the evaluation of Gn

c and the direct method for the CZM law calculationgive accurate and calibrated CZM law parameters for specific geometry andmaterial conditions.

Inverse Method

The inverse method consists on an iterative curve fitting procedure betweenexperimentally measured data and the respective FE predictions, considering aprecise description of the experimental geometry and approximate cohesive laws,established based on the typical behaviour of the material to be simulated (Baneaet al. 2011). The inverse characterization of adhesive bonds should be appliedindividually for each tested specimen to account for slight geometry variationsbetween specimens (Campilho et al. 2009a). By this technique, the value of Gn

c orGs

c, which corresponds to the steady-state value of Gn or Gs during crack propa-gation in the respective R-curve built from the fracture characterization test data, isinput in the FE model. To completely define the CZM law, approximate bulkvalues can be used for tn

0 or ts0 (Fig. 12) for the initiation of the trial and error

iterative process (Campilho 2009). Tuning of the cohesive parameters is performedby a few numerical iterations until an accurate prediction of the experimental datais achieved. Examples of reliable experimental data for the iterative fitting pro-cedure are the R-curve (Flinn et al. 1993), the crack opening profile (Mello andLiechti 2006), and more commonly the P–d curve (Li et al. 2005b).

In the work of Campilho (2009), an inverse fitting methodology was consideredfor the definition of a shear CZM law by the ENF test (continuum approach) for aductile epoxy adhesive bond (Hunstman Araldite� 2015) with tA = 0.2 mm.Notwithstanding the crack measurement difficulties arising from crack growth

Fig. 22 RepresentativeCZM law shapes in tensionloading for0.09 B tA B 1.0 mm (Ji et al.2010)


without opening (Blackman et al. 2005), the use of correction fluid along the crackgrowth path allowed a precise estimation of the current crack length by an imagingprocedure, consisting on image recording during the tests with 5 s intervals using a10 MPixel digital camera. Estimation of Gs

c was made possible by correlation ofP–d–a by the time elapsed from the test initiation. The testing time of each P–ddata point was calculated from the current value of d and the loading rate. Thecorrespondence to the values of a was established knowing the testing time of eachpicture. Gs

c was estimated by three methods: the CCM, the corrected beam theory(CBT) and the compliance-based beam method (CBBM), this last one notrequiring the measurement of a during crack propagation as it computes Gs

c onlyusing the experimental compliance. Data reduction by the three methods for fivetested specimens showed similar CCM and CBBM results, and smaller values ofCBT. The FE analysis faithfully represented each specimen geometry and mea-sured value of a0. A user defined continuum-based trapezoidal CZM formulation(Campilho et al. 2008a), coupled to Abaqus�, was considered to account for theadhesive ductility. Despite this fact, in terms of spatial modelling, the cohesiveelements had zero thickness (Fig. 23).

The CBBM values of Gsc, corresponding to the steady-state value of the

respective R-curves, were used as input in the FE models. The remaining cohesiveparameters (ts

0 and dss) were estimated by fitting the experimental and numerical

P–d curves of each specimen. Figure 24 shows the experimental and FE P–dcurves for one tested specimen after the fitting procedure.

Figure 25 shows the average shear CZM law and respective values of Gsc (JIIc),

ts0 (ru,II), ds

s (d2,II) and dsf (du,II), and also the CZM laws range after individual

application of the inverse fitting principles to five experimental P–d curves.The manual fitting allowed a clear insight on the influence of the CZM

parameters on the FE P–d curves shape. Gsc, which is the input value in the

simulations, mainly influences the peak load. Higher values of ts0 increase the peak

load and the specimen stiffness up this value, leading to a more abrupt post-peakload reduction. Finally, ds

s plays an important role on the roundness of the P–dcurve near the peak value. These findings indicated that a unique solution for theshear CZM law of the adhesive could be guaranteed by the inverse technique.

Fig. 23 Deformed shape of the ENF specimen during propagation, with boundary and loadingconditions (Campilho 2009)


Lee et al. (2010) proposed a systematic procedure to estimate the local CZMparameters of an adhesive bond, using single-leg bending (SLB) mixed modetests with tension or shear as the dominant modes, for the extrapolation of thepure mode laws, thus simplifying the inverse fitting technique. SLB specimenswith co-cured adherends (upper carbon fibre reinforced composite adherend andlower steel adherend) were tested under different mode mixities, due to theallowance of controlling this parameter by changing the specimens geometry,for validation of the obtained pure tension and shear data. All fractured speci-mens showed a cohesive failure of the bond, with evidence of adhesive andfibres on the bonding surface of the steel adherends. Measurement of Gc wascarried out from the test data, and the mode mixity was defined from theclassical beam theory. Linear extrapolation of the measured data for both tensionand shear mode dominant tests allowed the definition Gn

c and Gsc, as depicted in

Fig. 26. The mode decompositions resulted in Gnc = 140 N/m and Gs

c = 280N/m for pure tension and shear, respectively, within the range of typical valuesfor epoxy matrices.

A mixed mode CZM was considered for the reproduction of the test results,built using triangular CZM laws for pure tension and shear, and whose criteria forstress softening and complete separation are consistent with those of Eqs. 6 and 7,

Fig. 24 Experimental andFE P–d curves comparisonfor one tested specimen(Campilho 2009)

Fig. 25 Average shear CZMlaw and deviation afterapplication of the inversefitting principles to fivespecimens (Campilho 2009)


respectively. Definition of the missing cohesive parameters (Knn, Kss, tn0 and ts

0)used the design of experiments (DoE) and kriging metamodel (KM) techniques.

The DoE was built considering the aforementioned parameters to be determinedas the DoE design variables, and setting the variable levels and required samplingrates for the experiments by the central composite method. As a result of theanalysis, the quantity of 50 simulations was considered as the minimum number ofexperiments to obtain the four parameters. The KM, by allowing a mathematicalbasis between the system inputs (sampling points of the cohesive parameters) andoutputs (errors between the simulation results and experimental data), allowsattaining an optimized solution by minimization of the error ( f – y in Fig. 27).

An error function was defined as the load difference between the FE andexperimental P–d curves, at different pre-established values of d within the testrange (the vertical dashed line in Fig. 27 relates to the limit of the evaluationregion). The CZM parameters to be defined in pure tension and shear were then

Fig. 26 Linear extrapolationof Gn

c and Gsc from the mixed

mode test results (Lee et al.2010)

Fig. 27 Error defined by theKM, by comparison of the FEand experimental data(Lee et al. 2010)


estimated by a nonlinear optimization algorithm, the sequential quadratic pro-gramming algorithm, for minimization of the experimental/FE deviation.The obtained values of CZM parameters in each pure mode of loading wereconsidered to built the pure tension and shear CZM laws, and were regarded as theoptimized values to describe the co-cured SLB tests under the mixed mode tensionor shear dominant modes, although applicable to pure modes or different modemixities. Validation of the proposed CZM laws was carried out by consideration oftwo intermediate mode mixities (modification of the composite adherend thicknessto 1.28 and 1.48 mm), by comparison of the test results and FE simulations withthe previously defined parameters for the dominant modes mixities. Comparison ofthe FE and test P–d curves is provided in Fig. 28, giving maximum errors of nearly5% for both modelling conditions.

The mode mixities for the two reported conditions were superimposed on theplot of Fig. 26, further reinforcing the linear extrapolation assumption of Gn

c andGs

c, as the new data fitted perfectly on the proposed linear relationship. From theglobal results, the authors concluded that the proposed procedure accuratelydescribed the fracture behaviour of mixed mode joints, showing the advantage ofnon-requirement of two separate tests as it is usually performed (e.g., DCB fortension and ENF for shear characterization).

Static Applications of Cohesive Zone Models

The majority of CZM studies relate to static applications. Concerning bondedjoints, Yang et al. (1999) used a CZM, within the continuum framework, tosimulate a pure tensile fracture in bonded DCB and T-peel joints, not specifyingthe adhesive used. A trapezoidal CZM law was employed to simulate the adhesivebond behaviour (Fig. 29).

The authors concluded that Gnc (C0 in Fig. 29) and tn

0 (rc in Fig. 29) are thedominant parameters in the FE analyses, while the two shape parameters dn

0/dnf

Fig. 28 P–d curves for SLB specimens with varying mode mixities: composite adherendthickness of 1.28 (a) and 1.48 mm (b) (Lee et al. 2010)


(d1/dc in Fig. 29) and dns /dn

f (d2/dc in Fig. 29) have a marginal influence on theresults. As a result, the shape parameters were arbitrarily fixed at dn

0/dnf = 0.15 and

dns /dn

f = 0.5, while Gnc and tn

0 were fitted by an inverse method between theexperimental and FE data of the DCB specimens. By testing the proposed CZMlaw at varying conditions, the authors assumed that it could be used to predict thetensile fracture of different test geometries.

The same authors considered an identical trapezoidal CZM law (Fig. 29) forelastic–plastic crack growth modelling in shear by the continuum approach of atA = 0.4 mm ductile adhesive bond of Ciba� XD4600, considering the ENF test(Yang et al. 2001). The value of Gs

c was determined with a trial and error inverseprocedure, by comparing the experimental and FE data for one particular set ofdimensions. The two shape parameters, ds

0/dsf and ds

s/dsf, were arbitrarily equalled to

0.15 and 0.5, respectively, because of numerical evidence that their effect on thefracture process was not significant.

Fig. 29 Trapezoidaltraction–separation law intension of Yang et al. (1999)

Fig. 30 Experimental andnumerical load–deflectionplot for the ENF test (Yanget al. 2001)


ts0 was estimated by the assumption that in the ENF tests performed, the local

strains at the crack tip were approximately 40%. Using the shear stress–shearstrain (s – c) experimental curve of the adhesive obtained by torsion tests ofadhesively-bonded butt joints, the value of ts

0 = 35 MPa was found for c = 40%,and thus it was used in the subsequent analyses. Fitting of the experimental and FEP–d curves for one specimen is presented in Fig. 30. Results from a FEM cal-culation using only the continuum properties of the adhesive bond with no failurecriterion were superimposed in Fig. 30, showing that the deformation behaviour ofthe specimen is only accurately captured until crack initiation, rendering the resultsinvalid for crack propagation. The obtained values of Gs

c and ts0 were used to

simulate fracture in ENF specimens with different values of tP, and an excellentcorrelation to equivalent experiments was found. Therefore, the proposed CZMlaw was considered appropriate for the selected value of tA and strain rate of theENF tests. The authors also emphasized on the large difference of the CZMparameters for tension and shear loadings, justifying the use of mode dependentCZM relations to analyze the mixed mode fracture of bonded structures.

Yang and Thouless (2001) simulated the mixed mode fracture of plasticallydeforming asymmetric T-peel specimens and single lap joints using a mode-dependent CZM to simulate the adhesive bond (Ciba� XD4600), equally by acontinuum modelling approach. Tension and shear fracture parameters obtainedfrom previous works were combined with mixed mode propagation criteria to pro-vide quantitative predictions of the deformation and fracture load of bonded struc-tures under a mixed mode loading. Tension and shear CZM laws, previouslydeveloped for this adhesive in the works of Yang et al. (1999), (2001), respectively,were used without modification with an energetic propagation criterion similar tothat of Eq. 7. Since experimentally both evaluated geometries undergo adhesivefailures, the shear CZM parameters from the work of Yang et al. (2001) were used,since in this work adhesive failures also occurred in the ENF specimens tested.However, tension CZM parameters from the work of Yang et al. (1999) were notconsidered to be reliable, as they related to cohesive failures. Consequently, thesewere determined by the wedge test on adhesively bonded DCB specimens. Interfacialcrack growth was induced placing a Teflon� strip prior to specimen curing.

For the T-peel specimens, the FE results matched well the experiments in termsof r–d curve, strains and rotations, extent of fracture and asymmetry of bending(Fig. 31). For the single lap joints, the FE simulations accurately predicted theload-deformation behaviour of the joints, capturing the influence of tP on thefracture loads and strains. All relevant features of the deformed shapes, loads,displacements and crack propagation were well captured by the simulations.

Kafkalidis and Thouless (2002) performed a FE analysis of symmetric andasymmetric single lap joints using a continuum CZM approach that included theadhesive plasticity by means of trapezoidal CZM laws. The CZM laws, propa-gation criterion and analysis procedure were similar to those of Yang and Thouless(2001). Using CZM parameters determined for the particular combination ofmaterials used, the FE predictions for different bonded shapes (varying the overlaplength and tP) showed excellent agreement with the experimental observations.


Fig. 31 Comparison between the FE predictions and experimental observations of anasymmetrical T-peel specimen (Yang and Thouless 2001)

Fig. 32 Experimental andFE single lap joint strength asa function of the overlaplength for symmetricspecimens (Kafkalidis andThouless 2002)


Figure 32 compares the measured and FE peak load values for a series of sym-metric single lap joints with tP = 2.0 mm aluminium adherends bonded with atA = 0.25 mm bond of Ciba� XD4600.

The slight discrepancies on the predicted strengths of Fig. 32 were justified bythe sensitivity of the joint behaviour to the boundary conditions. In fact, theexperimental configuration results in a slight misalignment of the grips, which wasnot accounted for in the simulations. Figure 33 shows the FE predictions for thesequence of deformation and crack growth in an asymmetric joint with a tP ratio oftwo, which was consistent with the experimental behaviour. Overall, the FE modelspredicted accurately the failure loads, displacements and deformations of the joints.

A few parametric studies on bonded joints or repairs are also available that useCZM’s for optimization of the geometry (Campilho et al. 2009b), or evaluation ofgeometrical modifications to the adherends or adhesive bond to increase their loadbearing capabilities (Campilho et al. 2008b). The analysis of Karac et al. (2011)extends the typical applications of CZM’s to rate dependent simulations (localapproach), by using an implicit finite volume method coupled to the CZM analysis.In the work of Crocombe et al. (2006), a local CZM with a triangular shape wasdeveloped to predict the strength of environmentally degraded adhesively bondedsingle lap joints. Apart from the CZM parameters, the model included twomoisture dependent fracture parameters, all of these experimentally calibrated bymixed-mode flexure (MMF) tests. The mechanical diffusion events were alsoimplemented within Abaqus�, resulting on overall accurate predictions of thedegraded strength of the joints.

Campilho et al. (2009a) experimentally tested carbon-epoxy scarf repairs intension, using scarf angles between 2 and 458, considering CZM’s for the strengthprediction. To account for the ductile behaviour of the adhesive (HuntsmanAraldite� 2015), a trapezoidal CZM including the adhesive plasticity was used.The cohesive laws of the adhesive layer—continuum approach—and compositeinterlaminar and composite intralaminar (in the transverse and fibre direction)—local approach—necessary to replicate numerically the experimental failure paths,were previously characterized with DCB and ENF tests for tension and shear,respectively, using an inverse methodology. Validation of the proposed model

Fig. 33 FE deformationsequence and crack growth inan asymmetric single lap joint(Kafkalidis and Thouless2002)


with experimental data was accomplished in terms of global stiffness, strength andcorresponding value of d, and failure mode. The layout of cohesive elements in themodels is depicted in Fig. 34.

The adhesive representative elements were inserted along the scarf replacingthe adhesive bond, the interlaminar elements were positioned between differentoriented plies, the transverse intralaminar elements were used vertically in the 908plies to simulate intralaminar matrix cracking, and the fibre elements were placedvertically in the 08 plies to simulate fibre cracking. Two distinct failure modeswere experimentally observed: types A and B failures. Type A failure wasobserved for the repairs with 158, 258 and 458 scarf angles, consisting on acohesive failure of the adhesive. Type B failure occurred for the scarf angles of 28,38, 68 and 98, representing a mixed cohesive and interlaminar/intralaminar failure(Fig. 35). A detailed discussion for this failure mode modification was provided,based on a stress analysis of the repairs. The CZM simulations accuratelyreproduced the experimental fracture modes, and the failure mode modification

Fig. 34 Layout of cohesive elements with different CZM laws in the scarf repair FE models(Campilho et al. 2009a)

Fig. 35 Type B failure for a 38 scarf angle repair: experimental fracture surfaces (a) and FEprediction (distorted image for a clear visualization) (b) (Campilho et al. 2009a)


between types A and B, showing the effectiveness of CZM modelling for staticapplications.

Figure 36 reports the strength results, showing an exponential increase with thereduction of the scarf angle, related to the increase of available shear area for loadtransfer between the adherends and the patch, accompanied by a reduction of peelstresses along the scarf length.

Generally, experimental and FE results showed a good agreement on the globalelastic stiffness, strength and respective value of d, although the latter exhibitedsome variations, which was imputed to difficulties on the attainment of the selectedvalue of tA during the fabrication of the specimens. On account of the describedresults, the authors concluded that CZM’s can be successfully applied to predictthe mechanical behaviour of bonded structures.

The FE study of Campilho et al. (2009c) addressed the tensile strength opti-mization of carbon-epoxy single and double strap bonded repairs by geometricmodifications at the overlap. CZM’s were considered for local modelling ofthe adhesive, i.e., to provide crack growth, whilst the ductile adhesive bond(Huntsman Araldite� 420) was modelled by solid FE elements. Validation of theprocedure was initially accomplished with experimental results from the literature.The following techniques and respective combinations were tested: chamfering thepatch outer and inner faces at the overlap ends, plug filling the adherends gap withadhesive, use of fillets of different shapes and dimensions at the patch ends, andchamfering of the outer and inner adherend edges. For single strap repairs, the 458fillet (accounting for the entire patch thickness), and adherend inner and outer

Fig. 36 Experimental and FE strength of the scarf repairs as a function of the scarf angle(Campilho et al. 2009a)

Fig. 37 Single strap repair combining a 458 fillet with adherend inner and outer chamfering(Campilho et al. 2009c)


chamfering were the most effective solutions. By the combination of these threemodifications (Fig. 37), an overall strength improvement of nearly 27% wasattained.

For the double strap repairs, a 12% strength improvement was achieved byconsideration of a 458 fillet and plug filling the adherends gap. Parametric studieswere also performed regarding the adherend/patch lay-ups and the adherend gapeffects on the predicted strength. For the single strap geometry, the adherends lay-up revealed a particular significance because of its influence on the repairstransverse flexure. Lay-ups with load-oriented plies at the adherend free surfaceswere recommended on account of the reduction of peel stresses. Figure 38a reportsto stiff and b to compliant adherends under a similar applied load, showing crackgrowth only for the compliant adherends repair because of higher peel stresses.

The double strap results showed a negligible perturbation on strength becauseof the adherends flexure elimination. On the other hand, the repairs were onlyaffected by the adherends gap for values smaller than 5 mm. A progressivestrength reduction was found below this value because of localized stress con-centrations at the overlap edges. In the end, design principles were proposed forthe strength improvement of adhesively bonded repairs on composite structures.

Fatigue Applications of Cohesive Zone Models

A large number of analytical and FE predictive techniques for the fatigue life inbonded joints have been proposed over the years. However, each one of thesesuffers from specific limitations in their functionality and applicability (Shenoyet al. 2010a). The fatigue strength prediction with CZM’s is a recent possibility,under intense investigation in the last 5 years (Munoz et al. 2006; Turon et al.2007a). Fatigue CZM’s can be a valuable tool for the design of bonded jointsby the safe-life prediction under various loading conditions. Implementation of

Fig. 38 Deformed configuration of the repairs for stiff (a) and compliant adherends (b) for thesame value of prescribed load (Campilho et al. 2009c)


fatigue CZM’s in FE softwares is also time and cost efficient, enabling designers toreduce experimental testing to a minimum (Khoramishad et al. 2010). Importantissues possible to account for with fatigue CZM’s are the accurate prediction ofcyclic damage initiation, and the inclusion of fatigue life-time prediction, becauseunder cyclic loads either one of these phases can be dominant depending on thespecimens geometry, materials, load range and testing/loading conditions.However, the main advantage of CZM’s for modelling fatigue crack growth is thatdamage initiates within un-cracked structures when the cyclic cohesive stressesexceed a pre-determined critical value (Moroni and Pirondi 2011).

With this purpose, the static formulation of CZM’s, generically described in‘‘Triangular Cohesive Zone Model’’, is adjusted to account for fatigue damage. Infact, for cyclic loadings, damage is not only dependent on the relative separations,dn and ds, but also on the load cycle count. An associated limitation to somefatigue CZM formulations is the geometry dependency, which restrict the appli-cability of the models to specific test specimens (Khoramishad et al. 2010).Regarding the CZM approach, continuum modelling is often selected for fatigueand, thus, this information will be neglected in the following references.

A few lines of analysis for fatigue applications are currently under develop-ment: (1) a cycle-by-cycle analysis for the damage progression (Roe andSiegmund 2003; Abdul-Baqi et al. 2005; Maiti and Geubelle 2005), (2) cyclicextrapolation techniques (Turon et al. 2007b) and (3) analysis based on themaximum fatigue load (Robinson et al. 2005; Tumino and Cappello 2007). Thesecond and third approaches are less demanding in which regards to computationalresources, whilst the former allows a more precise damage evolution to beimplemented, predicting more accurately features such as crack growth retardationafter overload (Roe and Siegmund 2003), although not being viable for high cyclefatigue (Khoramishad et al. 2010).

The cycle-by-cycle fatigue CZM (i.e., 1st approach) proposed by Maiti andGeubelle (2005) for crack growth in polymeric materials was upgraded from astatic CZM identical to that of ‘‘Triangular Cohesive Zone Model’’, considering anevolution law relating the cohesive stiffness, the rate of crack opening displace-ment and the cyclic count from the onset of failure. The fatigue component reliedon two parameters that were easily defined from the Paris curve, between the crackgrowth per cycle and the range of applied stress intensity factor. A different routewas undertaken by Roe and Siegmund (2003), which simulated the crack growth atan interface by a fatigue CZM that incorporated a cycle-by-cycle simulation (thus,still within the scope of the described 1st approach) where damage was incre-mented according to the stress level attained in the previous cycle. In these twoworks, the main handicap was the fatigue parameters calibration, performed byexperiment/FE comparisons, added to the aforementioned limitations related to thelimited applicability to different geometries and high cycle fatigue.

The proposed fatigue CZM of Turon et al. (2007b), following the seconddescribed line of analysis for fatigue crack modelling of composites, does notrequire calibration of the fatigue parameters, solely requiring the static CZMparameters and the Paris-like law coefficients from fatigue crack growth tests.


However, the proposed formulation is restricted to test geometries in which G canbe computed analytically and it is independent of a during the test, such as in theDCB test.

The work of Moroni and Pirondi (2011) followed the same principle tosimulate fatigue crack growth in bonded joints with aluminium adherends (theadhesive was not specified). A fatigue CZM was implemented by the authors inAbaqus� based on the static formulation of ‘‘Triangular Cohesive Zone Model’’,more specifically in Eq. 6, for damage nucleation, whilst the Benzeggagh–Kenane (1996) criterion was considered for damage propagation, instead of theenergetic criterion of Eq. 7. The crack growth rate was evaluated and imple-mented in the CZM with three Paris-like power laws proposed in the literature,but all of them expressed in terms of the current value of G: (1) the Kenane andBenzeggagh theory (Benzeggagh and Kenane 1996), in which the crack growthlaw parameters depend on the mixed mode ratio by two constitutive equations,(2) the model of Quaresimin and Ricotta (2006), requiring an equivalentG parameter, defined as the driving parameter for crack growth and making thecrack growth law parameters independent of the mode mixity (consequently,they could be defined from a pure mode fatigue crack growth test and subse-quently applied to varying mode mixity conditions) and (3) a simplification ofthe Wahab et al. (2002) model, which established the crack growth rate as afunction of the tension and shear components (each one defined by the respectivepure mode fatigue parameters).

The fatigue CZM algorithm (Fig. 39) was implemented in Abaqus� by user subroutines for each crack growth rate criteria. The cyclic load was initially divided inincrements with a finite number of cycles. For each increment (defined as incre-ment j), the number of accumulated cycles from the beginning of the analysis isexpressed as Nj, and the respective damage for each integration point underanalysis (i) is equal to Di

j. After, an incremental damage within the analysisincrement (DDi

j) is established for each integration point from the smaller valuebetween the increment needed to reach D = 1 (failure) and a user-defined valueDDmax.

Fig. 39 Schematic representation of the algorithm used for the crack growth simulation (Moroniand Pirondi 2011)


DD ji ¼ DDmax if 1� D j

i [ DDmax

DD ji ¼ 1� D j

i if 1� D ji \DDmax

: ð8Þ

Concurrently with DDij, the user subroutine calculates DG as the J-integral over

a path around the cohesive zone. To represent a cyclic loading between a minimumload Fmin and a maximum load Fmax, the simulation is initially analysed with Fmax

and the respective maximum applied strain energy release rate Gmax is defined.The value of DG is obtained using R = Fmin/Fmax by

DG ¼ 1� R2�

Gmax: ð9Þ

The number of cycles for the current increment, DNij, is selected from the

minimum value of all CZM element integration points (DNminj ). The number of

cycles (Nj ? 1) and the damage distribution (Dj ? 1) are finally updated and theanalysis jumps to the next increment. The described model was tested by differentgeometries: in pure tension by the DCB specimen, in pure shear by the ELS test,and under mixed mode loading with a mixed mode ELS (MMELS) test. For thepure mode tests, the a versus cycle count predictions were extremely accurate andindependent of the Paris-like law employed.

As for the mixed mode results by the MMELS test, the crack growth ratecalculations differs between models (Fig. 40 shows results for model (1)).However, for all models the implemented algorithm was able to correctly modelthe crack growth prediction reported by the reference analytical models.

Robinson et al. (2005) followed the third line of analysis, by implementation ofa cumulative damage model based on the maximum fatigue load within a staticCZM. However, for the proposed formulation, the fatigue parameters were notindependent of the mode mixity. As a result, these parameters required calibrationfor each testing conditions. A couple of years later, Tumino and Cappello (2007)partially solved this limitation by making the fatigue parameters dependent on themode mixity. However, the proposed model became quite complicated because of

Fig. 40 CZM prediction andanalytical comparison ofa versus cycle count for theMMELS test and Paris-likelaw (1) (Moroni and Pirondi2011)


the numerous parameters of the fatigue model and cumbersome calibrationprocess.

Khoramishad et al. (2010) developed a fatigue CZM for adhesively bondedjoints with 2024-T3 aluminium adherends and the adhesive FM� 73 M OST(Cytec). The model was based on the maximum fatigue loading conditions (thirdline of analysis described) that are independent of the joint geometry, relying onlyon the adhesive system. Triangular damage laws were considered, and a strain-based fatigue damage model was developed and coupled to a static CZM forsimulation of the fatigue influence on the structures behaviour. Two differentconfigurations, bonded with FM� 73 M OST (Cytec) toughened epoxy filmadhesive and with the same adhesive system, were experimentally tested—singlelap joints in tension and laminated doublers in bending—giving different stressstates and mode mixities. The development and calibration of the proposed fatiguemodel were carried out for the single lap joint tests, which was subsequentlyapplied without any modification to the doublers in bending. Static tests wereinitially performed on both geometries for the definition of the static CZMparameters. The static CZM was tuned by an iterative technique to match theexperimental responses, similar to that described in ‘‘Inverse Method’’. Whilst thestatic CZM principles were identical to those described in ‘‘Triangular CohesiveZone Model’’, different criteria were considered for damage initiation and prop-agation: contrarily to Eq. 6, damage onset was predicted when either tn or tsexceeded the critical value (tn

0 or ts0, respectively); contrasting to the linear ener-

getic criterion for crack growth depicted in Eq. 7, the Benzeggagh–Kenane cri-terion was used, where g represents a material property.

Gcn þ Gc

s � Gcn

� :

Gs

Gn þ Gs

� �g

¼ Gn þ Gs: ð10Þ

Fatigue tests were subsequently carried out at 5 Hz and R = 0.1. For theassessment of damage evolution in the adhesive bond during fatigue loading, theback-face strain technique and in situ-microscopy were used. The back-face straindata was afterwards used for validation of the CZM by comparison to the predictedback-face strains. Fatigue damage simulation was implemented by degradation ofthe CZM response. A fatigue damage parameter was established, which evolvedduring the cyclic loading, based on the following damage evolution law

DD

DN¼ a emax � ethð Þb; emax [ eth

0; emax� eth

�; ð11Þ

with

emax ¼en

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffien

2

� �2þ es

2

� �2r

: ð12Þ

DD is the increment of damage, DN is the cycle increment, emax is the maxi-mum principal strain in the cohesive element (combination of the normal and shear


components of strain, en and es), eth is the threshold strain (a critical value of emax

below which no fatigue damage occurred) and a and b are material constants.Calibration against the test data was required for the parameters eth, a and b. Thedefined number of fatigue cycles in each increment resulted in fatigue damage,which was established by the numerical integration of the cyclic damage rate ofEq. 11, depending on the cyclic count of the current increment and emax. Asspecified in this equation, fatigue damage only occurs if emax [ eth.

Implementation of the fatigue damage was carried out by artificial degradationof the CZM response through the reduction of tn

0 and ts0 (Fig. 41). Fatigue crack

growth was divided into two steps: (1) a static CZM analysis was carried outconsidering the maximum fatigue load, using the static damage variable DS andproviding the stress/strain state at the beginning of the fatigue test, and emax wascalculated for all cohesive elements (Eq. 12); and (2) a cyclic damage variable(DF; Fig. 41) was established at each integration point of the cohesive elements,which was numerically integrated according to Eq. 11 in each increment, corre-sponding to a specified value of DN. The values of tn

0, ts0, Gn

c and Gsc were then

linearly reduced proportionally to DF (Fig. 42). The increment-by-incrementimplementation of this process, with the individual calculation of e and DF untilthe structure no longer sustains the maximum fatigue load, gave the predictedfatigue life. An iterative data fitting approach was undertaken for the single lapjoint results to determine appropriate fatigue model parameter values for theparticular adhesive system selected for the tests.

Validation was performed against the bonded doublers in bending (Fig. 43),giving an accurate correlation to the experimental data. Further validation was

Fig. 41 Fatigue degradationof cohesive elementproperties (Khoramishadet al. 2010)

Fig. 42 Fatigue damage(DF) and traction–separationresponse damage (DS)parameters (Khoramishadet al. 2010)


accomplished comparing the FE back-face strains with the test measurements fromthe undamaged state up to complete fatigue failure, equally giving a precise match.

2.4.2 Alternative Approaches to Cohesive Zone Modelling

This section describes ACDM techniques, i.e., other than CZM modelling, whichcan be categorized within the continuum framework if modelling finite volumes ofmaterial. In these methodologies, a damage parameter is established to modify theconstitutive response of materials by the depreciation of stiffness or strength, e.g.for thin adhesive bonds (Khoramishad et al. 2010), or composite delaminations ormatrix failure (Daudeville and Ladeveze 1993), to represent the severity ofmaterial damage during loading. This state variable can be used in a damageevolution law to model both pre-cracking damage uptake and crack growth. Thedamage variables can be categorized under two main groups: (1) variables thatpredict the amount of damage by the redefinition of the material constitutiveproperties but that do not directly relate to the damage mechanism, and(2) variables linked to the physical definition of a specific kind of damage, such asporosities or relative area of micro-cavities (Voyiadjis and Kattan, 2005). Thesecond group for damage variables is based on macroscopic material propertieswhose evolution is governed by a state equation. In all cases, different damagemechanisms concurrently occurring in a material can be accounted for in themodels, each of them by an independent damage variable. By ACDM, the growthof damage is defined as a function of the load for static modelling (Raghavan2005) or the cyclic count for fatigue analyses (Wahab et al. 2001; Imanaka et al.2003). Generic ACDM models for bulk materials are available in the literature(Lemaitre and Desmorat 2005), and respective modifications for particular kinds of

Fig. 43 Load-life fatiguedata and respective FEpredictions for the single laptests and doublers in bending(Khoramishad et al. 2010)


damage modelling, e.g. damage nucleation from voids and the formation of microcracks (Gurson 1977; Tvergaard and Needleman 1984; Kattan and Voyiadjis2005). ACDM’s have also been used to numerically model scenarios of constantand variable amplitude fatigue (Bhattacharya and Ellingwood 1998). For bondedjoints, little work is published in this field. Compared to fatigue CZM’s, ACDMtechniques do not provide a clear distinction between fatigue initiation andpropagation phases, although they can give a basis for the predictive analysis(Khoramishad et al. 2010). Nonetheless, the evolution of damage prior to macro-crack growth can be simulated. On the other hand, damage modelling with fatigueCZM’s is restricted to pre-defined crack paths and, on specific applications,ACDM’s may be recommended if the damage is more widespread or the failurepath is not known (Shenoy et al. 2010b).

Static Applications of Alternate Continuum Damage Mechanics

A few works are currently available using phenomenological ACDM techniquesfor static failure of structures, by the inclusion of one or more state variablesaffecting the macroscopic constitutive response of materials up to failure (Lemaitreand Chaboche 1985; Lemaitre 1986). The work of Sampaio et al. (2004) focusedon the static strength prediction of butt adhesive joints of aluminium adherendsbonded with a thin epoxy adhesive bond (Dow� DH331) by an analytical ACDMmodel accounting for the value of tA. The model combined a relatively straight-forward mathematical background with the possibility to describe complex non-linear mechanical behaviour of bonded joints, including strain softening and sizeeffects. The model was built on the assumption of two elastic bars (aluminiumadherends) and a damageable bar to model the adhesive bond, considering anapproximation of the 3D fracture behaviour to a 1D and equivalent ACDM modelwith a one-material homogeneous bar. An energy based formulation was estab-lished as a function of the axial strain, e, and an auxiliary damage variable, D, wasassociated with the dissipative mechanism of failure to account for the reduction ofelastic energy. To account for tA effects in bonded joints, a quadratic energyfunction was defined based on previous experimental evidence. For validation ofthe model, the predicted values of failure stress were plotted against experimentaldata for different values of tA, giving accurate representations of the tests and adecreasing strength of the butt joints with tA up to a bulk representativeconstant value.

Hua et al. (2008) proposed a mesh-independent ACDM model for strengthprediction of adhesively bonded joints under environmental degradation. Jointsbonded with a ductile adhesive (Hysol� EA9321) were studied for different sce-narios of environmental degradation. This was performed by introducing a dis-placement-based damage parameter into the constitutive equation of damagedmaterials that was moisture-dependent, calibrated using the MMF test on agedspecimens. The damage parameter allowed a linear function of the materialresponse in the absence of damage to be established, giving a depreciated value of


the actual stresses, depending on the equivalent plastic displacement rather thanstrain, to ensure mesh independent results. The implementation of this conceptrequired the definition of an equivalent length, linked to each solid elementintegration point. The equivalent plastic displacement was calculated from theenergy to rupture as the fracture work of the yield stress after the onset of damage.

The ACDM used by Hua et al. (2008) is currently implemented in Abaqus�

(Abaqus� Documentation 2009), and it also included the required mechanical-diffusion models and the elastic–plastic response of the adhesive and adherends,both of these obtained from bulk tensile tests. For each one of the aging conditions,the respective damage parameter was then directly applied to model failure insingle lap joints with aluminium and carbon-epoxy adherends, bonded with thesame adhesive. A von Mises yield model was considered for the adhesive bond.The joint strength predictions and the respective damage propagation patternmatched well the experimental data, showing that the developed ACDM provideda means of predicting environmental degradation in ductile bonded joints, wherefailure is predominantly within the adhesive bond.

Chen et al. (2011b) used the average plastic strain energy to predict crackinitiation and propagation in single lap joints, and also the failure load by anACDM technique. Two adhesives were used, Ciba� MY750 (diglycidyl ether ofbisphenol A) with hardeners HY906 (anhydride) and the same adhesive rubber-toughened with carboxyl terminated butadieneacryl (CTBN). These adhesives aredesignated by MY750 and CTBN, respectively, in the present description. Thisphysical fracture process was simulated by a FE analysis as follows. When aconverged solution was obtained after each applied load increment, a check wasmade to see whether the failure condition had been attained anywhere in the joint.Under these conditions, the values of E and Poisson’s ratio (m) of the material

Fig. 44 Failure progress forMY750 adhesive/steel singlelap joints; (a) schematicfailure step by step obtainednumerically; (b) crack patchobserved in the experiments(Chen et al. 2011b)


within the failed region were reset to zero (or near zero), such that it could deformalmost freely without transferring any load. The stresses within this region werealso reset to zero and then the whole system of equations (stiffness) was reas-sembled and solved. This FE model thus accounted for damage by voiding thecracked elements. All the checking and property depreciation were implemented inthe FE program and done automatically. A new load increment was not applieduntil the damage process had stopped.

The comparison between experimental and FE predicted failure loads showedthat the specific energy criterion used was fairly successful for fracture prediction,as shown in the example of Fig. 44, related to the MY750. All the FE fractureanalyses were based on the most realistic conditions: all considered the largedisplacements theory; all materials were treated elasto-plastically based on theirexperimental r–e curve. The experimental failure loads compared very well withthe experimental results with or without a fillet of adhesive as shown in Table 1 forthe case of a ductile adhesive (CTBN).

The work of Liu and Wang (2007) presented an experimental and ACDM studyof 3D double strap bonded repairs under a tensile load, using carbon-epoxylaminates and patches. A [(0/90/ ± 45/90/0)2]S lay-up was considered for thelaminate, and several quasi-isotropic lay-ups were tested for the patches. An epoxy

Table 1 Failure loads (in kiloNewton) for CTBN (ductile adhesive) bonded joints(Chen et al. 2011b)

Case Exp. failure load Standard deviation Numerical failure load Difference Error (%)

cst 9.51 0.53 9.54 0.03 0.3csftc 11.03 0.52 10.44 -0.59 -5.3clt 11.80 0.25 12.00 0.20 1.7clftc3 12.51 0.41 12.45 -0.06 -0.5clftc1 15.65 1.51 15.00 -0.65 -4.2cbt 8.76 0.73 10.08 1.32 15.1cbftc 11.08 1.19 10.80 -0.28 -2.5

cst steel without fillet; csftc steel with fillet; clt 2L73 aluminium without fillet; clftc3 2L73 withdefective fillet interface; clftc1 2L73 aluminium with small defect in fillet; cbt 2024 TBaluminium without fillet; cbftc 2024 TB aluminium with fillet

Fig. 45 Progressive damagealgorithm of Liu and Wang(2007)


adhesive film with tA = 0.12 mm was used. The implemented ACDM was basedon the degradation of the continuum element properties when specified criteriawere locally satisfied at a specified element. Different criteria were considered: (1)the Tsai-Wu criterion to predict fibre breakage and matrix cracking of the laminateand patches, (2) the Ye delamination criterion to detect delamination between pliesand (3) the maximum shear stress criterion for adhesive bond failure. In theACDM algorithm (Fig. 45), the analysis began with all material properties set totheir initial values. Then, the failure criteria were applied to the structure in eachincremental load step.

When each one of the criteria was attained at a specific element, the contri-bution of each stress component towards the failure index was assessed to identifywhich one contributes the most, allowing defining the dominant damage mecha-nism. The corresponding elastic properties, either of the composite or adhesive,were depreciated by multiplication with the stiffness reduction parameter, k, whichwas previously estimated by extensive comparative studies, since it largelyinfluences the strength and failure mechanism predicted by the ACDM. After thedegradation procedure, the analysis then restarted without increment of load untilno further damage was found in that step, or final failure of the structure wasattained. A convergence study defined the optimal increment of load, corre-sponding to the best match between accuracy of the strength predictions andcomputational time to attain the solution.

Validation considered double strap repairs with different values of patchthickness and patch diameter, and three failure mode possibilities were introduced.In mode A, failure initiates in the adhesive bond, leading to patch debonding.The laminates then fractures transversely near the hole, causing catastrophicfailure. In mode B, the patches fail to sustain the transmitted loads by the structureand split transversely. Almost simultaneously, the laminates fracture in anidentical manner. Mode C failure is characterized by damage initiation at thesingularity regions at the patch edges, growing to the laminate as transverse cracks.

Fig. 46 Predicted strengthversus experimental data ofthe double strap repairs (Liuand Wang 2007)


Figure 46 reports the good correspondence between the predicted strengths andthe experiments (groups 1–3 relate to patch thickness of 0.3, 0.7 and 0.875 mm,respectively; patch diameter of 40 mm for groups 1 and 2, and 50 mm for group 3).

Group 1 specimens fractured experimentally by mode B, which was accuratelycaptured by the numerical models (Fig. 47a), with almost simultaneous transversefractures of the patch and laminate. This was substantiated by the reduced value ofpatch thickness and its inability to reinforce the laminate near the hole. Fracturefor groups 2 and 3 corresponded to mode A, initiating in the adhesive bond, due topeel stress concentrations caused by the thicker patches. After losing the support ofthe patches, the laminate failed by transverse cracking initiating at the hole edges.This behaviour is consistent with the numerical analysis described earlier (Fig. 47bfor group 2). From these results, it was possible to conclude that for the materialand geometric parameters evaluated, the failure mode is ruled by the patchthickness. On the other hand, group 3 revealed the highest strength, justified by the

Fig. 47 ACDM failure process for group 1 (a) and group 2 (b) in two load stages (Liu and Wang2007)


larger patch diameter. Since the average strength of groups 1 and 2 was similar, itis concluded that this parameter was ruled by the patch diameter. After validationof the ACDM methodology, the influence of several geometric parameters on therepairs strength and failure mode was further studied (patch diameter, patchthickness, patch lay-up and tA), providing design principles for repairing bondedrepairs of composite structures.

Fatigue Applications of Alternate Continuum Damage Mechanics

Fatigue ACDM applications are not as widespread as static implementations(Solana et al. 2010), although a few formulations were proposed recently. Wahabet al. (2001) compared fracture mechanics and ACDM approaches for fatigue life-time prediction of bonded double lap joints with carbon-epoxy adherends, showingthat the proposed continuum model was more suited for constant amplitude fatiguethan fracture mechanics. Variable amplitude fatigue was not addressed in thiswork, but there is ample evidence of damage or crack growth accelerations ordecelerations occurring under variable amplitude due to load interaction effects,depending on the type of material (Shenoy et al. 2010b). Currently, scarce worksare published on the effect of variable amplitude fatigue on bonded joints byACDM.

Shenoy et al. (2010b) investigated the variable amplitude fatigue behaviour ofsingle lap joints by ACDM, made of 7075 T6 aluminium adherends bonded withthe epoxy adhesive film FM� 73 M (Cytec), considering various types of variableamplitude conditions. Significant load interaction effects were found by theapplication of the Palmgren–Miner law to the experimental data, showing that asmaller amount of fatigue cycles at higher fatigue loads gives a significantreduction in the fatigue life by accelerating damage growth. Two different lines ofanalysis were conducted for life prediction: fracture mechanics and ACDM. Twofracture mechanics-based techniques were tested, accounting for a different cyclecounting procedure. Although these did not include load interaction effects, whichcould over predict the fatigue life if these were found to accelerate damage, thefracture mechanics results consistently under predicted the joints fatigue life forthe entire range of fatigue spectra. This was because of the perceived inability offracture mechanics models to account for the crack initiation phase that showed tobe more dominant than the load interaction effects.

The ACDM approach used a relation between localised damage and plasticstrain in an empirical continuum damage law, allowing modelling the initiationand propagation phases of the joints fatigue life, and also accounting for loadhistory effects by the accumulation of incremental damage. The ACDM predictedthe fatigue life within the experimental scatter, oppositely to the fracturemechanics techniques. However, further development of the formulation wasproposed by the authors, concerning the form of the damage law used andrespective implementation within the FE software. Nonetheless, the comparativeanalysis to the experiments made clear that ACDM techniques have great potential


for predicting fatigue failure in bonded joints subjected to variable amplitudefatigue.

Shenoy et al. (2010a) presented a unified ACDM model to predict the fatiguefailure of bonded joints, wherein the evolution of fatigue damage in the adhesivewas defined by a power law function of the micro-plastic strain. Single lap jointswere considered with 7075 T6 aluminium adherends and the adhesive Cytec FM�

73 M, including in the FE models the fillet radius emerging from the fabricationprocedure. Fatigue testing was carried out at 5 Hz with R = 0.1. The ACDM wasimplemented by a sub routine in the MSC Marc FE software, aiming to propose auniversal methodology that could be widely applicable.

In the described formulation, a single damage evolution law was implementedto predict all the main parameters characterising the fatigue life of bonded joints,i.e., the progressive evolution of damage, crack initiation and propagation lives,back-face strains up to failure, and strength and stiffness degradation. The fatiguemodel is schematically described in Fig. 48. The ACDM algorithm requires asinput the material properties, joint geometry and boundary conditions, and a smallnumber of test results for the fatigue life are also needed to estimate the damageevolution law parameters. A number of user implemented algorithms were thencoupled to the FE analysis for the predictions. The rate of damage evolution wasassumed to be a power law function of the localised equivalent plastic strain, i.e.:

dD

dN¼ m1 ePð Þm2 ; ð13Þ

where D stands for the damage variable (D = 0 for undamaged material; D = 1for completely damaged material), N relates to the fatigue cyclic count, m1 and m2

Fig. 48 Schematic representation of the proposed fatigue model (Shenoy et al. 2010a)


are material parameters to be experimentally determined and eP is the localisedequivalent plastic strain. eP was chosen for damage progression in the proposedapproach to allow a means of inducing a set value of e below which the structuredoes not sustain damage (wherever eP = 0). If the applied value of load is not highenough, the adhesive bond endures eP = 0 and, hence, no fatigue damage isaccumulated because the load is below the endurance limit for the adhesive.Because of this, eP should be regarded as an endurance limit for a particular mesh,rather than an accurate measure of adhesive plasticity. Simulation of the damageevolution is performed by numerical integration of Eq. 13 over each element,which is followed by crack propagation when D = 1. By this algorithm, thenumber of cycles to failure can be calculated for different fatigue load conditions.A disadvantage of the methodology was related to the mesh dependency of theparameters m1 and m2, and necessity of adjustment for a particular mesh size,added to the requirement of optimization for the load range in which theseparameters could be applied. On the other hand, modelling possibilities includedthe straightforward implementation of a visco-elastic/plastic constitutive model forthe adhesive to simulate time dependent straining (potentially used to model thecreep enhanced fatigue failure of bonded joints), and, since damage is calculatedover every element, the non-uniformity of damage present in the adhesive can alsobe simulated.

For the fatigue life prediction, the load–life (L–N) curve was chosen, because inbonded joints the stress fields are extremely non-uniform and it is not possible todefine a relation between the average shear stresses in lap joints and the respectivemaximum stresses. The total fatigue life, Nf, can thus be separated into crackinitiation (Ni, number of cycles to crack onset) and propagation (Np, number ofcrack propagation cycles at failure) stages as follows

Nf ¼ Ni þ Np ð14Þ

By this analysis, it is possible to divide Nf into these two components, showingthe fatigue life proportion between crack initiation and propagation. For thestrength and stiffness degradation, an association was made with the damagewithin the adhesive (Eq. 13). Thus, the strength reduction of a bonded structureowing to fatigue damage could be easily calculated by application of an increasingload until complete failure. Actually, once the adhesive was damaged or cracked, aseries of increasing loads were applied until the model became unstable, indicatingthat the joint cannot bear further loads, thus giving an approximate strength for thejoint. The progressive damage modelling algorithm can be summarized in eightsteps as follows:

• Step 1 a FE model of the bonded structure is built and the values of N and D areset to zero for all elements of the adhesive bond,

• Step 2 a non-linear static analysis is performed and eP is calculated for all theelements in the adhesive bond,

• Step 3 a check is made on whether the analysis converges. If yes go to Step 4,otherwise N = Nf and the program is stopped,


• Step 4 the damage rate, dD/dN, is globally determined by Eq. 13,• Step 5 the updated value of D in each element is estimated using the damage rate

calculated in the previous step as

D ¼ Dþ dD

dN� dN; ð15Þ

where dN corresponds to the number of cycles of the current load increment,• Step 6 a check is made if D = 1. If yes, the respective element is deleted, and

the analysis jumps back to Step 2,• Step 7 if D = 1 in a given element the reduced element properties are calcu-

lated as

E ¼ E0 � 1� Dð Þ; ð16Þ

ryp ¼ ryp0 � 1� Dð Þ and ð17Þ

b ¼ b0 � 1� Dð Þ: ð18Þ

E0, ryp0 and b0 are the initial values of E, yield stress and plastic surface modifierconstant for the parabolic Mohr–Coulomb model, respectively, and• Step 8 the updated value of N is calculated and the analysis jumps to Step 2 and

it is repeated.

A convergence test was conducted to this algorithm for determination of theoptimum value of dN for each load increment, showing converged results fordN B 100, regarding the damage curve and the number of cycles to failure.Parameter calibration and ACDM model validation were carried out for single lapjoints. Figure 49 reports to crack growth prediction versus experimental resultscomparison for two values of maximum fatigue load and R = 0.1. For a maximumfatigue load of 7.5 kN, the predicted values of a were slightly bigger than the

Fig. 49 Crack growthprediction versusexperimental resultscomparison for two values ofmaximum fatigue load andR = 0.1 (Shenoy et al.2010a)


corresponding experiments, although with 6.5 kN the predictions are more accu-rate. For both scenarios the predicted crack growth rate was slightly higher thanthat of the experiments. Nonetheless, taking into account the elevated scattercharacteristic of fatigue life data, both predictions were satisfactory.

Figure 50 depicts the total fatigue life calculated by ACDM (addressed as UFMin the figure’s legend), showing a good correspondence with the experiments. Bydividing the total life into initiation and propagation phases as represented inEq. 14, evidence shows that the predicted proportion of initiation life increaseswith the reduction of fatigue load, which is entirely consistent with the currentexperiments and also with those from a previous work by the authors (Shenoyet al. 2009). The predicted crack propagation life using a conventional fracturemechanics approach, considering a Paris-like damage law, is superimposed to thedata of Fig. 50. The fracture mechanics approach underestimated the fatigue lifebecause it only predicted the propagation life. However, it was closer to the testdata at higher fatigue loads, where the proportion of initiation life to total life islower. The authors concluded that the proposed ACDM technique with properparameter calibration is accurate in predicting variable amplitude fatigue strength,with the advantage of quick adaption to combination with creep fatigue. As pre-viously discussed, changes in the nature of the damage law or damage parametermay be required for specific applications.

2.5 Extended Finite Element Method

The XFEM is a recent improvement of the FE method for modelling damagegrowth in structures. It uses damage laws for the prediction of fracture that arebased on the bulk strength of the materials for the initiation of damage and strainfor the assessment of failure (defined by Gn

c), rather than the values of tn0/ts

0 or dn0/ds

0

used for the CZM’s. XFEM gains an advantage over CZM modelling as it does not

Fig. 50 Experimental/ACDM comparison of theextended life versus N curve,showing crack initiation andtotal fatigue life; andconventional fracturemechanics prediction forcomparison (Shenoy et al.2010a)


require the crack to follow a predefined path. Actually, cracks are allowed to growfreely within a bulk region of a material without the requirement of the mesh tomatch the geometry of the discontinuities neither remeshing near the crack (Mo-hammadi 2008). This method is an extension to FE modelling, whose fundamentalfeatures were firstly presented in the late 90s by Belytschko and Black (1999). TheXFEM relies on the concept of partition of unity and it can be implemented in thetraditional FE by the introduction of local enrichment functions for the nodaldisplacements near the crack to allow its growth and separation between the crackfaces (Moës et al. 1999). Due to crack growth, the crack tip continuously changesits position and orientation depending on the loading conditions and structuregeometry, simultaneously to the creation of the necessary enrichment functions forthe nodal points of the finite elements around the crack path/tip.

Varying applications to this innovative technique were proposed to simulatedifferent engineering problems. Sukumar et al. (2000) updated the method to 3Ddamage simulation. Modelling of intersecting cracks with multiple branches,multiple holes and cracks emanating from holes were addressed by Daux et al.(2000). The problem of cohesive propagation of cracks in concrete structures wasstudied by Moës and Belytschko (2002), considering three-point bending and four-point shear scaled specimens. More advanced features such as plasticity, con-tacting between bodies and geometrical non-linearities, which show a particularrelevance for the simulation of fracture in structures, are already available withinthe scope of XFEM. The employment of plastic enrichments in XFEM modellingis accredited to Elguedj et al. (2006), which used a new enriched basis function tocapture the singular fields in elasto-plastic fracture mechanics. Modelling ofcontact by the XFEM was firstly introduced by Dolbow et al. (2000) and after-wards adapted to frictional contact by Khoei and Nikbakht (2006). Fagerström andLarsson (2006) implemented geometrical nonlinearities within the XFEM. Fatigueapplications for XFEM were proposed very recently (Xu and Yuan, 2009; Sabsabiet al. 2011), but these have not yet been applied to the mixed mode fracture ofbonded joints. As a result, only static applications of XFEM applied to bondedjoints will be considered.

2.5.1 Extended Finite Element Method Formulation

Although a few static implementations of the XFEM were developed in recentyears for scenarios other than bonded joints, the generic Abaqus� embeddedformulation will be described in this section (Abaqus� Documentation, 2009). TheXFEM considers an initial linear elastic behavior of the materials, which is rep-resented by an elastic constitutive matrix that relates stresses with the normal andshear separations of the cracked elements. Damage and failure are simulated inXFEM by suitable damage initiation criteria and damage laws between the realand phantom nodes of a cracked element (to be detailed further in this section).The damage initiation criteria can rely on the maximum principal stress or strain,while the traction–separation laws that simulate material degradation up to failure


can be linear or exponential. Damage initiation is assessed by Abaqus� typicallyby the maximum principal stresses or strains, calculated from the stress/strain stateat each integration point of the solid finite elements. The damage initiationfunction (f) can be expressed as (Abaqus� Documentation, 2009)

f ¼ rnh ir0

n

� �; ð19Þ

where rn relates to the maximum principal stress at an integration point and rn0 to

the material strength in tension. The Macaulay brackets interpretation is identicalto that of Eq. 6, i.e., it is used to specify that a purely compressive stress state doesnot induce damage. As an extension to the conventional FE technique, the XFEMis based on the integration of enrichment functions in the FE formulation, whichallows modelling the displacement jump between crack faces that occurs duringthe propagation of a crack. The fundamental expression of the displacement vectoru, including the displacements enrichment, is written as (Abaqus� Documentation2009)

u ¼XN

i¼1

Ni xð Þ ui þ H xð Þai þX4

a¼1

Fa xð Þbai

" #: ð20Þ

Ni(x) and ui relate to the conventional FE technique, corresponding to the nodalshape functions and nodal displacement vector linked to the continuous part of theformulation, respectively. The second term between brackets, H(x)ai, is only activein the nodes for which any relating shape function is cut by the crack and can beexpressed by the product of the nodal enriched degree of freedom vector includingthe mentioned nodes, ai, with the associated discontinuous shape function, H(x),across the crack surfaces (Abaqus� Documentation, 2009)

HðxÞ ¼ 1 if ðx� x�Þ � n� 0�1 otherwise

�: ð21Þ

x is a sample Gauss integration point, x* is the point of the crack closest to x,and n is the unit vector normal to the crack at x* (Fig. 51). Finally, the third termis only to be considered in nodes whose shape function support is cut by the cracktip and is given by the product of the nodal enriched degree of freedom vector of

Fig. 51 Representation ofnormal and tangentialcoordinates for an arbitrarycrack (Campilho et al. 2011d)


this set of nodes, bai ; and the associated elastic asymptotic crack-tip functions,

Fa(x) (Sukumar and Prevost 2003). Fa(x) are only used in Abaqus� for stationarycracks, which is not the current scenario.

In the presence of damage propagation, a different approach is undertaken,based on the establishment of phantom nodes that subdivide elements cut by acrack and simulate separation between the newly created sub elements. By thisapproach, the asymptotic functions are discarded, and only the displacement jumpis included in the formulation. Propagation of a crack along an arbitrary path ismade possible by the use of phantom nodes that initially have the exactly samecoordinates than the real nodes and that are completely constrained to the realnodes up to damage initiation.

In Fig. 52, the highlighted element has nodes n1–n4. After being crossed by acrack at UC, the element is partitioned in two sub-domains, XA and XB. The dis-continuity in the displacements is made possible by adding phantom nodes (ñ1–ñ4)superimposed to the original nodes. When an element cracks, each one of the twosub elements will be formed by real nodes (the ones corresponding to the crackedpart) and phantom nodes (the ones that no longer belong to the respective part of theoriginal element). These two elements that have fully independent displacementfields replace the original one, constituted by the nodes ñ1, ñ2, n3 and n4 (XA) and n1,n2, ñ3 and ñ4 (XB). From this point, each pair of real/phantom node of the crackedelement is allowed to separate according to a suitable cohesive law up to failure. Atthis stage, the real and phantom nodes are free to move unconstrained, simulatingcrack growth. In terms of damage initiation, Abaqus� allows the user to define initialcracks, but this is not mandatory. Regardless the choice taken, Abaqus� initiates andpropagates damage during the simulation at regions experiencing principal stressesand/or strains greater than the corresponding limiting values specified in the trac-tion–separation laws. Crack initiation/propagation will always take place orthogo-nally to the maximum principal stresses or strains.

2.5.2 Static Applications of the Extended Finite Element Method

XFEM applications to bonded joints are scarce and extremely recent in the liter-ature, and this section actually describes the entire range of applications found

Fig. 52 Damage propagation in XFEM using the phantom nodes concept: before (a) and afterpartitioning (b) of a cracked element into sub elements. (Campilho et al. 2011d)


while writing this work. Premchand and Sajikumar (2009) studied by the XFEMthe variation of the tensile and shear stress intensity factors in adhesively bondedjoints with different crack lengths, using the adhesive FM� 300-2 and aluminiumadherends. The XFEM model, coupled to the conventional FE method, wasimplemented in a Matlab� code, for the geometry of Fig. 53a, in which theadhesive holds a flaw with the depicted shape. A series of analyses were carriedout under tensile and shear loadings, giving estimations for stress intensity factorsin both modes of loading. Figure 53b relates to the joint deformed shape afterhorizontal crack propagation induced by a pure tensile loading applied at theadherend edges.

The study by Campilho et al. (2011d) checked the XFEM feasibility to modelcrack propagation and to predict the fracture behaviour of a thin bond of twostructural epoxy adhesives under tension by the DCB test using the XFEM, undervarying restraining conditions: stiff and compliant adherends. The damage laws ofthe XFEM were built by experimental determination of Gn

c and tn0, by DCB and bulk

tensile tests, respectively. Two DCB configurations were selected to check thesuitability of the XFEM in simulating fracture in bonded joints. Configuration Acorresponds to testing of a brittle epoxy adhesive (XN1244 by Nagase Chemtex)between stiff steel adherends, while in configuration B a ductile adhesive(Araldite� 2015) with compliant carbon-epoxy adherends was tested. Two methodswere employed to evaluate Gn

c: the CCM and the CBT, including the effects of cracktip rotation and deflection by using a correction factor, since the formulationassumes clamped adherends at the crack tip. The 2D XFEM analysis was performedin Abaqus� and the damage laws were assumed as triangular, i.e., with linearsoftening after tn

0, by using the properties obtained in the DCB and bulk tension tests.The steel and composite adherends were regarded as elastic isotropic and elastic

orthotropic, respectively, whilst the adhesive bond was modelled by the XFEM.

Fig. 53 Bonded joint geometry for stress intensity factor calculation by XFEM (a) andhorizontal crack propagation after a pure tensile load (b) (Premchand and Sajikumar 2009)


Validation of the XFEM and the proposed laws was accomplished by comparisonof the DCB experimental P–d curves with the output of the XFEM simulations forthe same geometry. Figure 54 depicts crack growth by the XFEM algorithm,initiating at the crack tip (a) and growing horizontally along the bondline (b) forconfiguration A. Figure 55 compares the experimental and XFEM P–d curves ofthe DCB specimens for configuration A, showing the classical concave shape afterthe peak load corresponding to crack growth at a constant Gn

c value, and anaccurate prediction of the specimens behaviour by using the XFEM with thepreviously characterized parameters.

For all conditions, the elastic stiffness, peak load, and load during propagationshowed a good agreement, which testified the suitability of XFEM to simulate bondedstructures for the conditions specified in the the analysis, i.e., pure tensile loading.

Campilho et al. (2011a) tested the CZM and XFEM formulations embedded inAbaqus� for the simulation of bonded single and double lap joints between alu-minium adherends and bonded with a brittle adhesive (Araldite� AV138). Overlaplengths between 5 and 20 mm were tested. The adhesive was characterized undertension and shear, which allowed the determination of the XFEM damageparameters. The 2D XFEM analysis considered geometrical non-linearities, withplane strain solid elements. Figure 56a shows the failure process for a single lapjoint with 20 mm overlap (detail at the overlap edge) using the principal strain

Fig. 54 Crack growth by the XFEM algorithm, initiating at the crack tip (a) and growinghorizontally along the bondline (b) for configuration A (Campilho et al. 2011d)

Fig. 55 Experimental andXFEM P–d curvescomparison for the DCBspecimens, consideringconfiguration A (Campilhoet al. 2011d)


criterion for the initiation of damage and estimation of crack growth direction. Atthis point, the direction of maximum strain led to propagation of damage towardsthe aluminium adherend.

When the crack front reached the adherend, damage propagated almost verti-cally due to the corresponding direction of principal strains at the crack tip(Fig. 56b), which clearly does not reflect the real behaviour of single lap joints.The authors concluded that damage propagation along the adhesive bond is thusrendered unfeasible with this technique, as it is currently implemented in Abaqus�,since the XFEM algorithm will always search for maximum stresses/strains at thecrack tip, shifting the crack to the adherends, disregarding what happens within theadhesive bond and thus preventing damage propagation along the bondline. As aresult of this handicap, a different solution was proposed by the authors, supportedby the brittleness of the adhesive used. The joint strength was estimated by theinitiation of cohesive cracking of the adhesive bond at the overlap edges, using themaximum principal strain criterion as it showed to be slightly less mesh sensitivethan the maximum principal stress equivalent.

Figure 57 compares the experimental and XFEM data considering the maxi-mum principal strain criterion, showing that the XFEM is moderately accurate insimulating these structures with brittle adhesives, which undergo a catastrophicfailure when the maximum strain of the adhesive is attained anywhere in thestructure. However, it was clarified that the chosen methodology was onlyacceptable due to the brittleness of the adhesive. If a ductile adhesive had beenused instead, the predictions would clearly underestimate the experiments.Figure 58 shows the results of the presented mesh dependency study, by plottingthe values of normalized peak load for element sizes at the overlap edges (equallength and height) between 0.05 and 0.2 mm, showing that, as expected, thepredictions are extremely mesh dependent.

Fig. 56 Progressive failure of a single lap joint with 20 mm overlap using XFEM (the arrowsrepresent the directions of maximum principal strain): damage initiation within the adhesive atthe overlap edges (a) and damage growth to the aluminium adherend (b) (Campilho et al. 2011a)


Globally, the direct comparisons between the experimental data and the outputof the simulations revealed fair predictions by the XFEM using the proposedsimplification to the original formulation. However, the XFEM did not show to besuited for damage propagation in bonded joints as it is currently implemented inAbaqus�, since the direction of crack growth is ruled by the maximum principalstresses/strains at the crack tip which, in bonded joints, typically leads to damagegrowth towards and within the adherends.

2.5.3 Suitability of the Extended Finite Element Methodfor Bonded Joints

The limitation of CZM’s is the requirement of the cohesive elements placementalong predefined paths where damage can occur, which can be hard to identify.The XFEM allows overcoming this limitation, since it propagates cracksarbitrarily within solid FE elements by using enrichment functions for the dis-placements. However, the current XFEM implementation in Abaqus� is restrictedto only one value of maximum strength/strain leading to the initiation of damage(by the maximum principal stress or strain criterion, respectively), which can be alimitation as damage in thin adhesive bonds is not consistent with that of bulkmaterials, due to the constraining effects imposed by the adherends (Xie and Waas2006). This does not allow the separation of the adhesive behaviour into the tensile

2000

4000

6000

8000

10000

12000

5 10 15 20

Pm

[N]

LO [mm]

Single-lap Experimental Single-lap Numerical (XFEM)

Double-lap Experimental Double-lap Numerical (XFEM)

Fig. 57 Experimental andXFEM strength comparisonas a function of the overlaplength (Campilho et al.2011a)

Pm

/Pm

av

g[%

]

Element length and height [mm]

140

120

100

80

60

40

0.05 0.1 0.15 0.2

Fig. 58 XFEM meshdependency study for thesingle lap joint with 20 mmoverlap (Campilho et al.2011a)


and shear, which can be in some cases mandatory for the accuracy of the results iflarge constraining effects are present in the bond (Campilho et al. 2009a). Thisclearly does not reflect the behaviour of bonded joints and can be attributed to analgorithm for propagation not still suited to multi material structures as it does notsearch for failure points outside the crack tip nor following the interfaces betweendifferent materials. Thus, the separation of the adhesive behaviour into the tensileand shear components that is performed for CZM’s is not possible to accomplish,which can be mandatory if large constraining effects exist (Campilho et al. 2009a).Apart from this feature, the current implementation of the method itself involvesan even more important handicap. It is known that, if no initial cracks are intro-duced in the models, the XFEM algorithm will automatically search for themaximum principal stresses/strains in each one of the structure materials (in thepresent scenario, in both the adhesive and adherends), to initiate damage propa-gation in the first locus in which these stresses/strains surpass the respectivematerial properties. During damage propagation, the XFEM algorithm continu-ously searches for the principal stress/strain direction at the crack tip, to specify thedirection of subsequent crack growth (Moës et al. 1999; Moës and Belytschko2002). For the specific case of single or double lap joints, cracking initiates in theadhesive bond orthogonally to the direction of principal stresses/stresses, growingup to the adhesive/adherend interface. Restriction of damage propagation only forthe adhesive bond is also rendered unfeasible to surpass this limitation as crackpropagation halts when the crack attains the aluminium (Campilho et al. 2011a).

As a result, under these conditions, slight inconsistencies between the predic-tions and experiments can occur. As a means to prevent this limitation of thecurrent XFEM implementation in Abaqus�, a modification that could account forthe distinction between tension and shear would increase the accuracy of thesimulations (Campilho et al. 2011d). Concerning the mesh dependency, the XFEMbehaves similarly to CZM’s, i.e., it is almost mesh independent for the simulationof fracture propagation, due to the fact that Gn

c is averaged over a finite region.Thus, for a large range of mesh sizes, provided that a minimum refinement is used,all the relevant features of the failure process are accurately captured (Campilhoet al. 2011a). However, the predictions of damage initiation load are meshdependent, since the state of stress at a region with large stress concentrationslargely depends on the mesh size. Apart from this, owing to the crack propagationmethod, i.e. orthogonally to the maximum stress/strain, mixed mode propagation isnot allowed, although it is frequent in bonded joints. This will invariably lead to awrong interpretation of the structures behaviour. Under these circumstances,assessment of the structures strength can be carried out by the initiation stress/strain criterion. However, this simplification is only expected to be sufficientlyaccurate for brittle adhesives that fail catastrophically when the peak stress/strainis achieved. For ductile adhesives, it will underestimate the structures strength, asthe adhesive will endure plasticization and redistribution of stresses before co-lapsing. Nonetheless, the failure load predictions, using the described simplifiedtechnique, are expected to be mesh size dependent because of the stress/straingradients at the sharp corners of bonded structures (Panigrahi and Pradhan 2007).


From this discussion it becomes clear that the XFEM, as it is currentlyimplemented, is only suitable for the identification of the locus of damage initi-ation in adhesive bonds, by comparing the maximum principal stress/strain in eachof the constituent materials to the respective maximum values. However, it doesnot show to be suited for the simulation of damage growth, as the principle fordefining the crack direction (orthogonal to the maximum principal stress/strain)does not model accurately the propagation of damage in multi-material structuresas it does not consider the initiation of damage outside the tip of the cracks thatemerge from the structure boundaries nor does it take into account the prospect ofdamage growth along interfaces between different materials. For the specific caseof bonded joints, a modification of the XFEM algorithm that would consider thesepossibilities would bring a significant breakthrough for the simulation of thesestructures, with the accuracy of CZM’s but eliminating the major handicap of thismethod to follow the damage paths specified by the placement of the cohesiveelements. Nonetheless, the XFEM is still a promising method that can be used in alarge number of applications for the strength prediction of structures.

3 Boundary Element Method

The BE method is another frequently used and now well-established numericalmethod, especially for dealing with fracture mechanics problems. A few consid-erations about the BE method are first given and then applications of the BEmethod to adhesive joints are commented.


Vable (2008) explains why the BE method can be useful for analysing adhesivejoints. Most of the methods of approximation convert a boundary value probleminto a set of algebraic equations. This is usually accomplished by representing thefield variable u (for example displacements in stress analysis) by a series

u ¼Xn

i

cifi; ð22Þ

where ci are the constants to be determined, and fi are an independent and completeset of n approximating functions. If there are no additional conditions on fi, thenthree types of error will be generated: error in the differential equation (ed) that isdistributed in the domain X shown in Fig. 59, error in natural boundary conditions(en) distributed on the boundary (Cn) where natural boundary conditions (condi-tions on traction for example) are imposed, and error in essential boundary con-ditions (ee) distributed on the boundary (Ce) where essential boundary conditions(conditions on displacements or temperature for example) are imposed.


All three errors ed, en, and ee are zero for an analytical solution. In approximatemethods, at least one of the error is not zero. This non-zero error must be mini-mized in some manner. The most general minimizing principle is the weightedresidue, in which the error is made orthogonal to a weighting function and can bewritten in the general form

ZX

Zwd

j eddx dyþZCe

wej eedsþ

ZCn

wnj ends ¼ 0; ð23Þ

where wdj ; we

j ; and wnj are the weighting functions chosen by some criteria that

depends upon the approximate method, ds is an element on C and x - y is thecoordinate system. In general, these weighting functions are related to theapproximating function fi through the operators of the differential equation orboundary conditions. On substituting the errors ed, en, and ee into Eq. 23, a linearsystem of algebraic equations in the unknown constants ci is obtained. In thedomain methods such as the FE and FD methods, the functions fi are chosen suchthat the error en or ee is forced to be zero. For example, ee is forced to zero in thestiffness version of FE method and en is forced to zero in the flexibility version ofFE method. In either case, the integral evaluation over X must be performed,requiring discretization of the domain. However, in boundary methods, such as BEmethod, the functions fi are chosen such that the error ed is forced to zero, leavingevaluation of integrals only over the boundary. Hence, in the BE method only theboundary needs to be discretized, as shown in Fig. 60. As a direct result, the size ofthe obtained system of equations can be much lower than in FE approaches.Meshing and remeshing is significantly simpler in the BE method than in the FEmethod (and other domain methods) for problems of shape optimisation or para-metric studies (rounding, tapering or shaping of the adherend or adhesive fillet), inwhich the boundary shape changes.

The discretization process converts the integrals in Eq. 23 to a sum of integralsover the elements. Thus, in the FE method and domain methods, when the domainintegral is set over the element to zero in Eq. 23, it is implied that the differentialequation is satisfied in an average sense over the element. But in the BE method this

Fig. 59 Homogeneousbody X


equation is satisfied exactly, as ed is forced to zero. If there are stress gradients in thedomain interior, then the BE method can potentially yield better resolution bychoosing points of stress evaluation very close to each other, while in domainmethods the resolution is dictated by the size of the elements. In the BE method,stresses are evaluated for example at the crack boundary and not at the integrationpoints inside the element, as for the FE method. For this reason, the BE method hasproven to have very good resolution of stress gradients for problems of stress con-centration and fracture mechanics. Therefore, it can potentially give good resolutionof stress gradients in the thickness direction of the adherend and the adhesive.However, if the stress gradients are along the boundary, as is the case on theadherend-adhesive interface, then the BE method like FE method requires a refinedmesh. One research challenge for the application of the BE method to bonded jointsis the development of a mesh refinement scheme for interface problems.

Another important feature is related to the treatment of infinite and semi-infiniteboundaries. In the BE method, the functions fi are the fundamental solutions of thedifferential equations, and therefore they implicitly satisfy the zero stress condi-tions at infinity. If there is a uniform stress at infinity then its value is simply addedto the integral equations. For this reason, in the BE method, only the boundary of ahole in an infinite medium needs to be discretised contrarily to domain methodsthat require the discretization of the exterior. However, joints have boundaries thatstart at the adhesive end and extend to infinity. Such boundaries will need spe-cialized infinite boundary elements.

Fig. 60 Comparisonbetween the general FEmethod and the BE methodapproach. a Structural model,b FE mesh, c BE mesh

3 Boundary Element Method 75

3.2 Application to Adhesive Joints

Given the above potential advantages, one would expect an extensive use of the BEmethod in the analysis of bonded joints. However, the reality is that a very limitednumber of studies concerning the use of the BE method can be found in theliterature. This is due to several disadvantages, described by Öchsner (2011). Firstof all, the derivation of the method is highly mathematical and the theoreticalfoundation is still not incorporated in many engineering degree courses. Thestiffness matrix in the case of the BE method might be much smaller compared tothe FE approach, but it is fully populated and many highly efficient equation solverswhich take advantage of the band structure obtained with FE approaches cannot beused. In addition, domain integrals must be considered in the case of non-linearities,which requires at least a coarse mesh inside the domain. Thus, the advantage of thereduced mesh on the boundary disappears. Typical applications of the BE methodin the context of adhesion technology are commonly found for the modelling ofcracks (fracture mechanics) and other types of stress singularities. It has beenwidely used to analyse metallic structures repaired by composites. For exampleYoung et al. (1988, 1992), Wen et al. (2002, 2003), Salgado and Aliabadi (1998),and Liu et al. (2009) used the BE method to analyse cracked sheets, in which thepatch and the cracked sheet were modelled using boundary elements and theadhesive was simulated as shear springs. However, these studies are focussed on thecracked plate rather than on the adhesive bond. Ikeda et al. (1998) combined the FEand BE methods to analyse a butt joint with a crack in the adhesive. The FE methodwas used in the adhesive and the adherends were modelled with the BE method.Computer power can be saved but the use of the BE method for adhesive modellingis limited. Lee (1998) investigated the singular stresses at the interface cornerbetween a viscoelastic adhesive and a rigid adherend subjected to a uniformtransverse tensile strain using the time-domain BE method. The model is presentedin Fig. 61. The BE analysis could capture the large stress gradients present at theinterface corner and the stress singularity. Cavallini et al. (2006) presented a BEtechnique for modelling an adhesively bonded structural joint. The formulationallowed the analysis of general lay-up laminates such as Glare material. A simplemodel consisting of a layer of adhesive on aluminium showed good agreement withan analytical model. Vable and Maddi (2006) addressed specific problems (i.e.,numerical modelling considerations which limited the application of the BEmethod in the past) related to bonded joints and BE simulation. In addition,numerical results of lap joints with isotropic elastic adherends and an elasticadhesive were presented, which demonstrated the potential of the BE method inanalysis of bonded joints. The effect of several spew angles was also studied. Goodagreements were found with a FE analysis. The authors pointed out that the BEmethod was a viable technique for stress analysis of bonded joints, since it had thepotential of producing good resolution of stress gradients and it is robust enough fora parametric study of joint parameters. Further improvements in the resolution ofstress gradients required development of a mesh refinement scheme for material


interfaces and development of infinite boundary elements for use with semi-ana-lytical integration schemes. The authors referred that the design of mechanicallyfastened joints was facilitated by stress concentration factor curves, which aredrawn as a function of various geometric parameters. If a similar design approach isto be developed with the concepts of stress intensity factors usage in bonded joints,then the BE method with its high accuracy and ease of modelling geometric featurescan well become a methodology of choice for stress analysis of bonded joints.

4 Finite Difference Method

The FD method is the first widely known approximation technique for partialdifferential equations. The method is described in detail in the works of Forsytheand Wasow (1960), Collatz (1966) and Mitchell and Griffiths (1980). A brief

Fig. 61 Analysis model of Lee (1998) for determination of interface stresses developed in avisco-elastic adhesive layer

3 Boundary Element Method 77

description of the FD method is firstly presented and then applications of the FDmethod to adhesive joints are given.


The FD method is a numerical technique for approximating the solutions to dif-ferential equations, by using FD equations to approximate derivatives. Physically,a derivative represents the rate of change of a physical quantity represented by afunction f(x) with respect to the change of its variable x. An example of a functionf(x) is graphically represented in Fig. 62. The differential df(x)/dx is defined by:

df xð Þdx¼ lim

Dx!0

Df

Dx� Df

Dx: ð24Þ

Equation 24 depicts the principle of finite difference. The smaller the stepsDx the closer the values between the differential and difference of the rate changeof the function. The two sources of error in the FD method are the round-off error(the loss of precision due to computer rounding of decimal quantities), and thetruncation error or discretization error, i.e., the difference between the exactsolution of the FD equation and the exact quantity.

To use a FD method to attempt to approximate the solution to a problem, onemust first discretize the problem’s domain. This is usually done by dividing thedomain into a uniform grid (Fig. 63). In the case of the FE method, the domainmust be discretized with elements (which are composed of nodes and bound-aries, and which may have a distorted shape), whereas the FD method requiresonly the discretization of the grid points, which are not joined to elements.Therefore, for the case of the FD method, the results are only obtained at thegrid points and not interpolated as in the case of the FE method. An interpolation

Fig. 62 Illustration of theapproximation by the FDmethod


between the grid points might be obtained with additional formulation. Öchsner(2011) refers that the major advantage of the FD method is the simple computerimplementation of the procedures. Therefore, in-house codes are easily imple-mented and new features are easy to add. The main disadvantages are theexpected complications for the simulation of boundary conditions for complexshaped geometries and the difficult attainment of symmetry for the stiffnessmatrix. For these reasons, the FD method is only applied to simple geometries.However, these problems may be solved using the FD energy method developedby Bushnell et al. (1971).

4.2 Application to Adhesive Joints

Application of the FD method to the analysis of adhesively bonded joints isrelatively limited due mainly to the availability of commercial FE programsthat offer a large range of features and enable to include complex geometriesand non-linear problems. For complex geometries and elaborate materialmodels, a FE analysis is preferable. However, for a fast and easy answer, aclosed-form analysis is more appropriate. Adhesive joints have been intensivelyinvestigated over the past 70 years and numerous analytical models have beenproposed (da Silva et al. 2009a, b). The first model (Fig. 64) proposed byVolkersen (1938) assumes that the adhesive deforms only in shear and that the

Fig. 63 Comparisonbetween the FE method andthe FD approach. a Structuralmodel; b FE mesh; c FD gridpoints

4 Finite Difference Method 79

adherends can deform in tension. The following partial differential equation isestablished

d2r1

dx2� k2r1 þ C0 ¼ 0; ð25Þ

where k2 ¼ Gata

1E1t1þ 1

E2t2

� �; C0 ¼ Ga

taP

bE2t2t1; Ga is the adhesive shear modulus

and E1 and E2 are the adherends Young’s modulus. This is a second order non-linear differential equation that has an algebraic solution of the form

r1 ¼ A cosh kxð Þ þ B sinh kxð Þ þ C0

k2 ; ð26Þ

where the two constants A and B are determined by the boundary conditions at theends of the overlap.

The pioneering analytical models of adhesively bonded joints such as thoseof Vokersen (1938) and Goland and Reissner (1944) are described by differ-ential equations that have an algebraic solution. However, many advancedadhesive joints analyses, such as those that include material nonlinearity(Bigwood and Crocombe 1990; Wang et al. 2003) and composite adherends(Srinivas 1975; Adams and Mallick 1992; Mortensen and Thomsen 2002),involve non-linear and non-homogeneous differential equations, and the solutionof these equations is often beyond the reach of classical methods. Numericalsolutions such as the FD method are then the only practical and viable ways tosolve these differential equations. For example, the Crocombe and Bigwood

Fig. 64 Single lap joint analysed by Volkersen (1938)


(1992) sandwich adherend-adhesive-adherend model can accommodate the non-linear stress response of both the adhesive and the adherends, and can also besubjected to several forms of loading. Numerical results were obtained by usinga FD method to solve a set of six non-linear first-order differential equations.Wang et al. (2003) extended the work of Crocombe and Bigwood (1992) toaccount for shear deformation in the adherend to predict adhesive failure inarbitrary joints subjected to large scale adherend yielding. They also obtained asystem of six non-linear first-order differential equations, which was solved bythe double (precision) boundary value problem finite difference (DBVPFD)solver. The results obtained were in close agreement with a FE analysis, exceptfor the localized stress and strain concentrations near the free edge. Additionalmodels that solved the governing equations using FD methods are those of Kimand Kedward (2001) and Xu and Li (2010). Kim and Kedward (2001) devel-oped a closed-form stress analysis of an adhesively bonded lap joint subjectedto spatially varying in-plane shear loading. The solution, while similar toVolkersen’s treatment of tension loaded lap joints, was inherently 2D and, ingeneral, predicted a multi-component adhesive shear stress state. Stresses inadhesively bonded composite tubular joints, subjected to torsion, were inves-tigated by Xu and Li (2010). The model resulted in a system of eighteensecond-order partial differential equations and six linear equations with twenty-four unknowns, which were solved by a mathematical model based on the FDmethod.

5 Conclusions

The most recent trends in numerical modelling of adhesive joints have been dis-cussed in this book. The three major numerical techniques, FE method, BE methodand FD method were described. The following conclusions can be drawn.

The FE method is by far the most common numerical tool. In the continuummechanics approach, the maximum values of stress, strain or strain energy, pre-dicted by the FE analyses, are typically used in the failure criterion where they arecompared with the corresponding material allowable values. However, it is knownthat these maximum predicted values are usually found very near to the singularpoints of the model (sharp corners or bi-material interfaces). Therefore, theirmagnitude strongly depends on how well the stress field around the singularity ismodelled (i.e., mesh refinement). Additionally, the maximum critical valuesobtained by the FE analyses are also dependent on the proximity of the criticalpoint from the stress or strain concentrator. In order to overcome this problem, acommon approach used by many researchers is to use the same variables (stress,strain or energy) but this time at some arbitrary distance from the point of sin-gularity, where the stress field is clear of any effects from the singular point. Thecritical distances must be usually calibrated from the FE results, and as a conse-quence, these critical values obtained can only be used for similar geometric and

4 Finite Difference Method 81

material configurations. There is usually no physical explanation relating thecritical distances with experimental observations. The average plastic densitybelongs to the same concept, but the use of this value gives a less sensitive result tothe mesh size used.

Continuum mechanics assumes that the structure and its material are continu-ous. Defects or two materials with re-entrant corners obviously are not consistentwith such an assumption. Cracks are the most common defects in structures, forwhich the method of fracture mechanics has been developed. Fracture mechanicscan be used to predict joint strength or residual strength if there is a crack tip or aknown and calibrated singularity. Fracture mechanics is more difficult to apply tostrength predictions for joints bonded with ductile adhesives, since Gc is notindependent of the joint geometry. This is mainly because the adherends restrictthe development of the yield zone in the adhesive bond or they can cause fractureof the adhesive if the material yields.

Damage mechanics has been used to model the progressive damage and failureof a pre-defined crack path, or arbitrarily within a structure. This is an emergingfield and the techniques for modelling damage can be divided into either local orcontinuum approaches. In the local approach, damage is confined to a zerothickness path (2D analyses) or surface (3D analyses). By the continuum approach,damage is modelled over a finite area (2D analyses) or volume (3D analyses).CZM’s, which can be categorized under these two lines of analysis, simulate themacroscopic damage along this path by the specification of a traction–separationresponse between paired nodes on either sides of pre-defined crack paths. In mostof the CZM’s, the traction–separation relations for the interfaces are such that,with increasing interfacial separation, the traction across the interface reaches amaximum (crack initiation), then it decreases (softening), and finally the crackpropagates, permitting a total de-bond. The whole failure response and crackpropagation can thus be modelled. A CZM simulates the fracture process,extending the concept of continuum mechanics by including a zone of disconti-nuity modelled by cohesive zones, thus using both strength and energy parametersto characterize the debonding process. This allows the approach to be of muchmore general utility than conventional fracture mechanics. The method is alsomesh insensitive, provided that enough integration points undergo softeningsimultaneously. It is similar to criteria where the stress or strain is averaged over afinite area so as to remove the singularity. Studies demonstrated that it is possibleto experimentally determine the appropriate cohesive zone parameters of anadhesive bond, and to incorporate them into FE analyses for excellent predictivecapabilities. However, CZM’s present a limitation, as it is necessary to knowbeforehand the critical zones where damage is prone to occur, and to place thecohesive elements accordingly. Also, for ductile materials, the shape of the trac-tion–separation law must be modified, which may give additional convergenceproblems.

The XFEM, expanding CZM by the allowance of crack propagation alongarbitrary directions within solid continuum elements, is not suited for damagepropagation in bonded joints as it is currently implemented in softwares, such as in


Abaqus�, since the direction of crack growth is ruled by the maximum principalstresses/strains at the crack tip which, in bonded joints, invariably leads to damagegrowth towards and within the adherends. This clearly does not reflect thebehaviour of bonded joints and can be attributed to an algorithm for propagationnot still suited to multi-material structures, as it does not search for failure pointsoutside the crack tip nor following the interfaces between different materials.Restriction of damage propagation only for the adhesive bond is also renderedunfeasible to surpass this limitation as crack propagation halts when the crackattains the aluminium. Nonetheless, studies were made with some simplificationhypotheses, which allowed fair approximations to the joints behaviour.

The BE method is still at an early stage of development concerning the analysisof adhesive joints. However, the inherent advantages of the method such as theeasy modelling of cracks, stress singularities and large stress gradients justify itsmore intensive use in the context of adhesive joints.

The FD method is useful for solving differential equations derived in complexanalytical models. It is generally used for own code development and solution ofnew analytical derivations for relatively simple geometries.

From the state-of-the-art description carried through in this book, it is clear thatnumerical methods have been largely exploited for some decades for the strengthand fracture prediction of bonded joints by varying approaches, and with formu-lations tailored for different load scenarios. In the near future, further develop-ments within this scope are expected, and a few research tendencies are proposedthat can effectively expand the use of these powerful tools. CZM’s will definitelybe improved, beginning from easier and more efficient calibration tools. Regardingthe analyses capabilities, further research on the residual strength of environ-mentally degraded joints is required, namely regarding degradation of the adhesiveand interfacial regions because of extreme conditions that structures can be sub-jected to in industrial applications. Rate dependent damage formulations are stillvery incipient and need exploiting, and fatigue CZM’s are very recent and needfurther validations and improvements for a widespread application to differentgeometries and load conditions. XFEM is also in the initial stages of developmentand it can be adjusted to be useful to bonded joints, if the aforementioned limi-tations of this method are surpassed.

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